Abstract Algebra Homework 3

import%20marimo%0A%0A__generated_with%20%3D%20%220.17.6%22%0Aapp%20%3D%20marimo.App()%0A%0A%0A%40app.cell%0Adef%20_()%3A%0A%20%20%20%20import%20marimo%20as%20mo%0A%20%20%20%20return%20(mo%2C)%0A%0A%0A%40app.cell%0Adef%20_(mo)%3A%0A%20%20%20%20nav_menu%20%3D%20mo.nav_menu(%0A%20%20%20%20%20%20%20%20%7B%0A%20%20%20%20%20%20%20%20%20%20%20%20%22%2Findex.html%22%3A%20f%22%7Bmo.icon('lucide%3Ahome')%7D%20Home%22%2C%0A%20%20%20%20%20%20%20%20%7D%2C%0A%20%20%20%20%20%20%20%20orientation%3D%22vertical%22%2C%0A%20%20%20%20)%0A%20%20%20%20nav_menu%0A%20%20%20%20return%0A%0A%0A%40app.cell%0Adef%20_(mo)%3A%0A%20%20%20%20mo.md(r%22%22%22%0A%20%20%20%20%23%20Spring%20Semester%20Homework%201%0A%20%20%20%20%22%22%22)%0A%20%20%20%20return%0A%0A%0A%40app.cell%0Adef%20_(mo)%3A%0A%20%20%20%20mo.md(r%22%22%22%0A%20%20%20%20%23%23%20P61.7%0A%0A%20%20%20%20**Let%20%24F%24%20be%20a%20field%2C%20%24V%20%3D%20F%5E2%24.%20We%20define%20an%20equivalence%20relation%20on%20%24V%20-%20%5Clbrace%200%20%5Crbrace%24%20by%20%24v%20%5Csim%20w%24%20if%20and%20only%20if%20%24v%20%3D%20%5Calpha%20w%24%20for%20some%20%24%5Calpha%20%5Cin%20F%5E%7B%5Ctimes%7D%24.%20The%20set%20of%20equivalence%20classes%20under%20this%20relation%20is%20called%20the%20one-dimensional%20projective%20space%20(or%20projective%20line)%20over%20%24F%24%20and%20is%20denoted%20by%20%24%5Cmathbb%7BP%7D%5E1(F)%24.%20We%20write%20elements%20of%20%24V%24%20as%20column%20vectors%2C%20and%20the%20element%20of%20%24%5Cmathbb%7BP%7D%5E1(F)%24%20that%20is%20the%20equivalence%20class%20of%20%24%5Cbegin%7Bpmatrix%7D%20x%20%5C%5C%20y%20%5Cend%7Bpmatrix%7D%24%20shall%20be%20written%20as%20%24%5Cbegin%7Bbmatrix%7D%20x%5C%5Cy%20%5Cend%7Bbmatrix%7D%24**.%0A%0A%20%20%20%20**Show%20that%20there%20is%20a%20well-defined%20action%20of%20%24PSL(2%2C%20F)%24%20on%20%24%5Cmathbb%7BP%7D%5E1(F)%24%20given%20by**%20%24%24%5Coverline%7B%5Cbegin%7Bpmatrix%7D%20a%26b%20%5C%5C%20c%26d%20%5Cend%7Bpmatrix%7D%7D%5Cbegin%7Bbmatrix%7D%20x%5C%5Cy%20%5Cend%7Bbmatrix%7D%20%3D%20%5Cbegin%7Bbmatrix%7D%20ax%2Bby%5C%5Ccx%2Bdy%20%5Cend%7Bbmatrix%7D%24%24%20**and%20that%20this%20action%20is%20doubly%20transitive%20where%20%24%5Coverline%7B%5Cbegin%7Bpmatrix%7D%20a%26b%20%5C%5C%20c%26d%20%5Cend%7Bpmatrix%7D%7D%24%20denotes%20the%20image%20of%20%24%5Cbegin%7Bpmatrix%7D%20a%26b%20%5C%5C%20c%26d%20%5Cend%7Bpmatrix%7D%20%5Cin%20SL(2%2CF)%24%20in%20%24PSL(2%2C%20F)%24**.%0A%0A%20%20%20%20Proof%0A%0A%20%20%20%20Follow%20the%20axioms%3A%0A%0A%20%20%20%201.%20If%20%24%5Coverline%7B%5Cbegin%7Bpmatrix%7D%20a%26b%20%5C%5C%20c%26d%20%5Cend%7Bpmatrix%7D%7D%20%3D%201%20%5Cin%20PSL(2%2CF)%24%20then%20%24b%20%3D%20c%20%3D%200%24%20and%20%24a%20%3D%20d%24%2C%20so%20%24%5Coverline%7B%5Cbegin%7Bpmatrix%7D%20a%26b%20%5C%5C%20c%26d%20%5Cend%7Bpmatrix%7D%7D%5Cbegin%7Bbmatrix%7D%20x%5C%5Cy%20%5Cend%7Bbmatrix%7D%24%20%24%3D%20%5Cbegin%7Bbmatrix%7D%20ax%5C%5Cay%20%5Cend%7Bbmatrix%7D%24%20%24%3D%20%5Cbegin%7Bbmatrix%7D%20x%5C%5Cy%20%5Cend%7Bbmatrix%7D%24%3B%0A%20%20%20%202.%20%24%5Coverline%7B%5Cbegin%7Bpmatrix%7D%20a'%26b'%20%5C%5C%20c'%26d'%20%5Cend%7Bpmatrix%7D%7D(%5Coverline%7B%5Cbegin%7Bpmatrix%7D%20a%26b%20%5C%5C%20c%26d%20%5Cend%7Bpmatrix%7D%7D%5Cbegin%7Bbmatrix%7D%20x%5C%5Cy%20%5Cend%7Bbmatrix%7D)%24%20%24%3D%5Coverline%7B%5Cbegin%7Bpmatrix%7D%20a'%26b'%20%5C%5C%20c'%26d'%20%5Cend%7Bpmatrix%7D%7D%5Cbegin%7Bbmatrix%7D%20ax%2Bby%5C%5Ccx%2Bdy%20%5Cend%7Bbmatrix%7D%24%20%24%3D%5Cbegin%7Bbmatrix%7D%20a'(ax%2Bby)%20%2B%20b'(cx%2Bdy)%5C%5Cc'(ax%2Bby)%2Bd'(cx%2Bdy)%20%5Cend%7Bbmatrix%7D%24%2C%20on%20the%20other%20hand%20%24(%5Coverline%7B%5Cbegin%7Bpmatrix%7D%20a'%26b'%20%5C%5C%20c'%26d'%20%5Cend%7Bpmatrix%7D%7D%5Coverline%7B%5Cbegin%7Bpmatrix%7D%20a%26b%20%5C%5C%20c%26d%20%5Cend%7Bpmatrix%7D%7D)%5Cbegin%7Bbmatrix%7D%20x%5C%5Cy%20%5Cend%7Bbmatrix%7D%24%20%24%3D%5Coverline%7B%5Cbegin%7Bpmatrix%7D%20a'a%2Bb'c%26a'b%2Bb'd%20%5C%5C%20c'a%2Bd'c%26c'b%2Bd'd%20%5Cend%7Bpmatrix%7D%7D%5Cbegin%7Bbmatrix%7D%20x%5C%5Cy%20%5Cend%7Bbmatrix%7D%24%20%24%3D%5Cbegin%7Bbmatrix%7D%20(a'a%2Bb'c)x%20%2B%20(a'b%2Bb'd)y%5C%5C(c'a%2Bd'c)x%2B(c'b%2Bd'd)y%20%5Cend%7Bbmatrix%7D%24%2C%20so%20they%20are%20the%20same.%0A%0A%20%20%20%20(the%20point%20in%20part%202%20is%20that%20the%20quotient%20image%20of%20product%20is%20the%20same%20as%20the%20product%20of%20quotient%20image%2C%20which%20is%20true%20since%20%24G%2FN%20%5Ctimes%20G%2FN%20%5Ccong%20(G%20%5Ctimes%20G)%2F(N%20%5Ctimes%20N)%24)%0A%0A%20%20%20%20For%20doubly%20transitivity%2C%20essentially%20for%20%24%5Cbegin%7Bbmatrix%7D%20x_1%5C%5Cy_1%20%5Cend%7Bbmatrix%7D%5Cne%5Cbegin%7Bbmatrix%7D%20x_3%5C%5Cy_3%20%5Cend%7Bbmatrix%7D%2C%5Cbegin%7Bbmatrix%7D%20x_2%5C%5Cy_2%20%5Cend%7Bbmatrix%7D%5Cne%5Cbegin%7Bbmatrix%7D%20x_4%5C%5Cy_4%20%5Cend%7Bbmatrix%7D%20%5Cin%5Cmathbb%7BP%7D%5E1(F)%24%20we%20want%20to%20find%20%24%5Coverline%7B%5Cbegin%7Bpmatrix%7D%20a%26b%20%5C%5C%20c%26d%20%5Cend%7Bpmatrix%7D%7D%24%20so%20that%20%24%5Cbegin%7Bcases%7D%20ax_1%2Bby_1%20%3D%20x_2%20%5C%5C%20cx_1%20%2B%20dy_1%20%3D%20y_2%20%5C%5C%20ax_3%2Bby_3%20%3D%20x_4%20%5C%5C%20cx_3%2Bdy_3%20%3D%20y_4%20%5Cend%7Bcases%7D%24.%20We%20can%20write%20this%20system%20as%20%24%5Cbegin%7Bpmatrix%7D%20x_1%26y_1%20%5C%5C%20x_3%26y_3%20%5Cend%7Bpmatrix%7D%5Cbegin%7Bpmatrix%7D%20a%26c%20%5C%5C%20b%26d%20%5Cend%7Bpmatrix%7D%3D%5Cbegin%7Bpmatrix%7D%20x_2%26y_2%20%5C%5C%20x_4%26y_4%20%5Cend%7Bpmatrix%7D%24.%20By%20assumption%20the%20determinant%20of%20%24%5Cbegin%7Bpmatrix%7D%20x_1%26y_1%20%5C%5C%20x_3%26y_3%20%5Cend%7Bpmatrix%7D%24%20(and%20%24%5Cbegin%7Bpmatrix%7D%20x_2%26y_2%20%5C%5C%20x_4%26y_4%20%5Cend%7Bpmatrix%7D%24)%20is%20never%20%240%24%20so%20we%20always%20have%20a%20solution.%0A%0A%20%20%20%20The%20only%20problem%20is%20that%20if%20we%20directly%20solve%20this%20system%20and%20%24%5Cdet%7B%5Cbegin%7Bpmatrix%7D%20x_1%26y_1%20%5C%5C%20x_3%26y_3%20%5Cend%7Bpmatrix%7D%7D%20%5Cne%20%5Cdet%7B%5Cbegin%7Bpmatrix%7D%20x_2%26y_2%20%5C%5C%20x_4%26y_4%20%5Cend%7Bpmatrix%7D%7D%24%20then%20%24%5Cdet%7B%5Cbegin%7Bpmatrix%7D%20a%26b%20%5C%5C%20c%26d%20%5Cend%7Bpmatrix%7D%7D%20%5Cne%201%24%20(determinant%20of%20product%20is%20product%20of%20determinant)%20so%20it%20is%20in%20%24PGL%24%20but%20not%20in%20%24PSL%24.%20In%20this%20case%2C%20say%20%24%5Cdet%7B%5Cbegin%7Bpmatrix%7D%20x_1%26y_1%20%5C%5C%20x_3%26y_3%20%5Cend%7Bpmatrix%7D%7D%20%3D%20a%20%5Cin%20F%24%20and%20%24%5Cdet%7B%5Cbegin%7Bpmatrix%7D%20x_2%26y_2%20%5C%5C%20x_4%26y_4%20%5Cend%7Bpmatrix%7D%7D%20%3D%20b%20%5Cin%20F%24%2C%20with%20%24a%5Cne%20b%24%2C%20then%20%24a%5E%7B-1%7Db%20%5Cin%20F%24%20(it%20is%20not%20%240%24%20because%20a%20field%20does%20not%20have%20zero%20divisor).%20Replace%20%24%5Cbegin%7Bpmatrix%7D%20x_1%5C%5Cy_1%20%5Cend%7Bpmatrix%7D%24%20by%20%24%5Cbegin%7Bpmatrix%7D%20a%5E%7B-1%7Dbx_1%5C%5Ca%5E%7B-1%7Dby_1%20%5Cend%7Bpmatrix%7D%24%20so%20it%20is%20still%20in%20the%20same%20equivalence%20class%2C%20then%20the%20solution%20will%20necessarily%20be%20in%20%24PSL%24%2C%20thus%20we%20have%20double%20transitivity.%0A%0A%20%20%20%20%23%23%20P61.9%0A%0A%20%20%20%20**Let%20%24F%24%20be%20the%20field%20of%20%247%24%20elements.%20For%20%240%20%5Cle%20i%20%5Cle%206%24%20let%20%24i%24%20represent%20%24%5Cbegin%7Bbmatrix%7D1%20%5C%5Ci%20%20%5Cend%7Bbmatrix%7D%20%5Cin%20%5Cmathbb%7BP%7D%5E1(F)%24%3B%20denote%20the%20remaining%20element%20%24%5Cbegin%7Bbmatrix%7D0%5C%5C1%20%20%5Cend%7Bbmatrix%7D%24%20of%20%24%5Cmathbb%7BP%7D%5E1(F)%24%20by%20%24%5Cinfty%24.%20View%20%24S_8%24%20as%20being%20the%20group%20of%20permutations%20of%20%24%5Clbrace%20%5Cinfty%2C0%2C%5Cdots%2C6%20%5Crbrace%24%20and%20let%20%24%5Cvarphi%3APSL(2%2C7)%20%5Cto%20S_8%24%20be%20the%20monomorphism%20arising%20as%20in%20Proposition%203.1%20from%20the%20action%20of%20%24PSL(2%2C7)%24%20on%20%24%5Cmathbb%7BP%7D%5E1(F)%24.%20Show%20that%20the%20image%20of%20%24%5Cvarphi%24%20is%20the%20subgroup%20of%20%24A_8%24%20generated%20by%20%24(0%20123456)%24%2C%20%24(142)(356)%24%20and%20%24(%5Cinfty%200)(13)(25)(46)%24**.%0A%0A%20%20%20%20Proof%0A%0A%20%20%20%20We%20know%20%24PSL(2%2C7)%24%20is%20simple%2C%20so%20the%20image%20of%20%24PSL(2%2C7)%24%20under%20%24%5Cvarphi%24%20would%20be%20a%20subgroup%20of%20%24A_8%24%20(because%20determinant%20%24%3D1%24%20corresponds%20to%20even%20permutations)%20that%20is%20isomorphic%20to%20%24PSL(2%2C7)%24.%0A%0A%20%20%20%20(The%20image%20of%20natural%20projection%20of)%20the%20matrix%20%24%5Cbegin%7Bpmatrix%7D%201%260%5C%5C1%261%20%5Cend%7Bpmatrix%7D%24%20is%20of%20course%20an%20element%20in%20%24PSL(2%2C7)%24.%20On%20the%20other%20hand%2C%20consider%20the%20(set-wise)%20stabilizer%20of%20%24%5Clbrace%200%2C%5Cinfty%20%5Crbrace%24%3A%0A%0A%20%20%20%20By%20some%20direct%20matrix%20multiplication%20calculation%2C%20if%20we%20want%20to%20map%20%240%20%5Cmapsto%200%24%20and%20%24%5Cinfty%20%5Cmapsto%20%5Cinfty%24%2C%20i.e.%20%24%5Cbegin%7Bpmatrix%7D%20a%26b%5C%5Cc%26d%20%5Cend%7Bpmatrix%7D%5Cbegin%7Bpmatrix%7D%201%5C%5C0%20%5Cend%7Bpmatrix%7D%20%3D%20%5Cbegin%7Bpmatrix%7D%20x%5C%5C0%20%5Cend%7Bpmatrix%7D%24%20and%20%24%5Cbegin%7Bpmatrix%7D%20a%26b%5C%5Cc%26d%20%5Cend%7Bpmatrix%7D%5Cbegin%7Bpmatrix%7D%200%5C%5C1%20%5Cend%7Bpmatrix%7D%20%3D%20%5Cbegin%7Bpmatrix%7D%200%5C%5Cy%20%5Cend%7Bpmatrix%7D%24%2C%20then%20%24b%24%20and%20%24c%24%20are%20forced%20to%20be%20%240%24%2C%20and%20%24a%2Cd%24%20are%20inverse%20of%20each%20other%20because%20the%20matrix%20has%20determinant%20%241%24%2C%20i.e.%20we%20need%20a%20matrix%20of%20the%20form%20%24%5Cbegin%7Bpmatrix%7D%20x%260%5C%5C0%26x%5E%7B-1%7D%20%5Cend%7Bpmatrix%7D%24%2C%20those%20matrices%20are%20generated%20by%20%24%5Cbegin%7Bpmatrix%7D%202%260%5C%5C0%264%20%5Cend%7Bpmatrix%7D%24%3B%20and%20if%20we%20want%20to%20map%20%240%20%5Cmapsto%20%5Cinfty%24%20and%20%24%5Cinfty%20%5Cmapsto%200%24%20then%20similarly%20we%20need%20a%20matrix%20of%20the%20form%20%24%5Cbegin%7Bpmatrix%7D%200%26-x%5E%7B-1%7D%5C%5Cx%260%20%5Cend%7Bpmatrix%7D%24%2C%20those%20are%20generated%20by%20%24%5Cbegin%7Bpmatrix%7D%200%263%5C%5C2%260%20%5Cend%7Bpmatrix%7D%24.%0A%0A%20%20%20%20Apparently%20%24%5Cbegin%7Bpmatrix%7D%201%260%5C%5C1%261%20%5Cend%7Bpmatrix%7D%24%20cannot%20be%20generated%20by%20the%20other%20two%20matrices.%20Since%20stabilizer%20of%20doubly%20transitive%20action%20is%20maximal%20(we%20proved%20this%20statement%20for%20stabilizer%20of%20single%20point%2C%20but%20it%20is%20really%20the%20same%20argument%20for%20the%20set-wise%20stabilizer)%2C%20the%20group%20%24%5Clangle%20%5Cbegin%7Bpmatrix%7D%201%260%5C%5C1%261%20%5Cend%7Bpmatrix%7D%2C%20%5Cbegin%7Bpmatrix%7D%202%260%5C%5C0%264%20%5Cend%7Bpmatrix%7D%2C%20%5Cbegin%7Bpmatrix%7D%200%263%5C%5C2%260%20%5Cend%7Bpmatrix%7D%20%5Crangle%24%20(under%20the%20natural%20projection)%20must%20be%20the%20whole%20%24PSL(2%2C7)%24.%20So%20the%20isomorphic%20image%20of%20%24PSL(2%2C7)%24%20inside%20%24A_8%24%20must%20be%20generated%20by%20the%20permutations%20induced%20by%20those%20three%20matrices.%0A%0A%20%20%20%20It%20is%20then%20not%20hard%20to%20verify%20they%20correspond%20to%20%24(0123456)%24%2C%20%24(124)(365)%24%20(it%20is%20the%20same%20as%20%24(142)(356)%24%2C%20just%20replace%20%24%5Cbegin%7Bpmatrix%7D%202%260%5C%5C0%264%20%5Cend%7Bpmatrix%7D%24%20with%20%24%5Cbegin%7Bpmatrix%7D%204%260%5C%5C0%262%20%5Cend%7Bpmatrix%7D%24)and%20%24(%5Cinfty%200)(13)(25)(46)%24%20as%20desired.%0A%0A%20%20%20%20%23%23%20P80.4%0A%0A%20%20%20%20**Let%20%24U%24%20be%20the%20subgroup%20of%20%24GL(n%2Cp)%24%20consisting%20of%20the%20upper%20uni-triangular%20matrices%2C%20and%20let%20%24Q%24%20be%20the%20subgroup%20of%20%24U%24%20consisting%20of%20all%20matrices%20whose%20%24(i%2Cj)%24%20entry%20is%20zero%20whenever%20%241%20%3C%20i%20%3C%20j%20%3C%20n%24.%20Determine%20%24Z(Q)%24%2C%20and%20show%20that%20%24Q%2FZ(Q)%24%20is%20abelian**.%0A%0A%20%20%20%20Proof%0A%0A%20%20%20%20By%20assumption%2C%20%24Q%24%20is%20the%20group%20such%20that%20a%20generic%20element%20looks%20like%20%24%24%5Cbegin%7Bpmatrix%7D%201%26a_%7B12%7D%26a_%7B13%7D%26%5Cdots%26a_%7B1~n-1%7D%26a_%7B1n%7D%20%5C%5C%200%261%260%26%5Cdots%260%26a_%7B2n%7D%20%5C%5C%200%260%261%26%5Cdots%260%26a_%7B3%20n%7D%5C%5C%20%5Cdots%26%5Cdots%26%5Cdots%26%5Cdots%26%5Cdots%26%5Cdots%5C%5C%200%260%260%26%5Cdots%261%26a_%7Bn-1~n%7D%5C%5C%200%260%260%26%5Cdots%260%261%5Cend%7Bpmatrix%7D%2C%24%24%20i.e.%20matrices%20with%20%241%24%20on%20the%20diagonal%2C%20arbitrary%20on%20the%20first%20row%20and%20last%20column%20(except%20for%20the%20top%20left%20and%20bottom%20right%20corner%2C%20of%20course)%2C%20and%20%240%24%20everywhere%20else.%0A%0A%20%20%20%20%24Z(Q)%24%20is%20by%20definition%20those%20%24Z%20%5Cin%20Q%24%20such%20that%20%24ZM%20%3D%20MZ%24%20for%20%24M%20%5Cin%20Q%24.%20Let's%20denote%20the%20terms%20of%20%24Z%24%20by%20%24z_%7Bij%7D%24%2C%20of%20%24M%24%20by%20%24m_%7Bij%7D%24%2C%20and%20just%20do%20some%20matrices%20multiplication.%20It%20is%20not%20very%20hard%20to%20see%20that%20(for%20a%20fixed%20%24Z%24)%3A%20%24ZM%20%3D%20MZ%24%20if%20and%20only%20if%20%24z_%7B12%7Dm_%7B2n%7D%20%2B%20z_%7B13%7Dm_%7B3n%7D%2B%5Cdots%20%2Bz_%7B1~n-1%7Dm_%7Bn-1~n%7D%24%20%24%3D%20m_%7B12%7Dz_%7B2n%7D%20%2B%20m_%7B13%7Dz_%7B3n%7D%2B%5Cdots%20%2Bm_%7B1~n-1%7Dz_%7Bn-1~n%7D%24%2C%20for%20**all**%20%24M%20%5Cin%20Q%24%2C%20so%20the%20only%20choice%20is%20to%20have%20all%20those%20%24z_%7Bij%7D%24's%20to%20be%20%240%24.%20The%20term%20we%20are%20not%20touching%20is%20%24z_%7B1n%7D%24%2C%20so%20a%20generic%20element%20of%20%24Z(Q)%24%20is%20a%20matrix%20with%20%241%24%20on%20the%20diagonal%2C%20arbitrary%20on%20the%20top%20right%20corner%2C%20and%20%240%24%20everywhere%20else.%0A%0A%20%20%20%20The%20center%20is%20always%20a%20normal%20subgroup%2C%20so%20the%20quotient%20is%20always%20a%20group%2C%20we%20just%20need%20to%20show%20it%20is%20abelian.%0A%0A%20%20%20%20Notice%20if%20we%20multiply%20a%20matrix%20by%20such%20%24Z%24%2C%20we%20merely%20add%20%24z_%7B1n%7D%24%20to%20the%20top%20right%20corner%20of%20the%20original%20matrix%20%24%24%5Cbegin%7Bpmatrix%7D%201%26a_%7B12%7D%26a_%7B13%7D%26%5Cdots%26a_%7B1~n-1%7D%26%5Cboxed%7Ba_%7B1n%7D%7D%20%5C%5C%200%261%260%26%5Cdots%260%26a_%7B2n%7D%20%5C%5C%200%260%261%26%5Cdots%260%26a_%7B3%20n%7D%5C%5C%20%5Cdots%26%5Cdots%26%5Cdots%26%5Cdots%26%5Cdots%26%5Cdots%5C%5C%200%260%260%26%5Cdots%261%26a_%7Bn-1~n%7D%5C%5C%200%260%260%26%5Cdots%260%261%5Cend%7Bpmatrix%7D%20%5Cmapsto%20%5Cbegin%7Bpmatrix%7D%201%26a_%7B12%7D%26a_%7B13%7D%26%5Cdots%26a_%7B1~n-1%7D%26%5Cboxed%7Ba_%7B1n%7D%2Bz_%7B1n%7D%7D%20%5C%5C%200%261%260%26%5Cdots%260%26a_%7B2n%7D%20%5C%5C%200%260%261%26%5Cdots%260%26a_%7B3%20n%7D%5C%5C%20%5Cdots%26%5Cdots%26%5Cdots%26%5Cdots%26%5Cdots%26%5Cdots%5C%5C%200%260%260%26%5Cdots%261%26a_%7Bn-1~n%7D%5C%5C%200%260%260%26%5Cdots%260%261%5Cend%7Bpmatrix%7D.%24%24%20It%20means%20each%20coset%20in%20the%20quotient%20consists%20precisely%20all%20matrices%20that%20only%20differ%20at%20the%20top%20right%20corner%2C%20and%20we%20may%20choose%20to%20use%20the%20matrices%20with%20%240%24%20at%20the%20top%20right%20corner%20to%20represent%20each%20coset.%0A%0A%20%20%20%20Now%20if%20we%20multiply%20%24A%2C%20B%20%5Cin%20Q%24%20in%20two%20ways%20%24A%20B%24%20and%20%24B%20A%24%2C%20it%20is%20not%20hard%20to%20see%20the%20result%20differs%20only%20possibly%20at%20the%20top%20right%20corner%2C%20but%20we%20chose%20to%20force%20the%20top%20right%20corner%20to%20be%20zero%2C%20so%20the%20result%20is%20always%20the%20same%20in%20the%20quotient%20group%2C%20and%20thus%20the%20group%20is%20abelian%20as%20desired.%0A%0A%20%20%20%20%23%23%20P80.6%0A%0A%20%20%20%20**Let%20%24U%24%20be%20the%20subgroup%20of%20%24GL(3%2C%20p)%24%20consisting%20of%20the%20upper%20uni-triangular%20matrices.%20Show%20that%20if%20%24p%24%20is%20an%20odd%20prime%2C%20then%20%24U%24%20is%20a%20non-abelian%20group%20of%20order%20%24p%5E3%24%20having%20no%20elements%20of%20order%20%24p%5E2%24.%20If%20%24p%20%3D%202%24%2C%20with%20which%20group%20of%20order%20%248%24%20is%20%24U%24%20isomorphic**%3F%0A%0A%20%20%20%20Proof%0A%0A%20%20%20%20%24U%20%3D%20%5Clbrace%20%5Cbegin%7Bpmatrix%7D%201%26a%26b%5C%5C0%261%26c%5C%5C0%260%261%5Cend%7Bpmatrix%7D%20%7C%20a%2Cb%2Cc%20%5Cin%20%5Cmathbb%7BZ%7D_p%20%5Crbrace%24%2C%20so%20%24%7C%20U%20%7C%20%3D%20p%5E3%24%20because%20each%20%24a%2Cb%2Cc%24%20can%20be%20chose%20from%20%24p%24%20different%20elements.%20Consider%20the%20matrices%20%24A%20%3D%20%5Cbegin%7Bpmatrix%7D%201%261%260%20%5C%5C%200%261%260%20%5C%5C%200%260%261%20%5Cend%7Bpmatrix%7D%24%20and%20%24B%20%3D%20%5Cbegin%7Bpmatrix%7D%201%260%260%5C%5C0%261%261%5C%5C0%260%261%20%5Cend%7Bpmatrix%7D%24%2C%20then%20%24A%2C%20B%20%5Cin%20U%24%20and%20we%20can%20calculate%20%24AB%20%3D%20%5Cbegin%7Bpmatrix%7D%201%261%261%5C%5C0%261%261%5C%5C0%260%261%20%5Cend%7Bpmatrix%7D%24%20%24%5Cne%20%5Cbegin%7Bpmatrix%7D%201%261%260%5C%5C0%261%261%20%5C%5C0%260%261%20%5Cend%7Bpmatrix%7D%3DBA%24%2C%20so%20%24U%24%20is%20not%20abelian.%0A%0A%20%20%20%20Moreover%2C%20if%20%24p%24%20is%20odd%2C%20a%20direct%20calculation%20gives%20%24%5Cbegin%7Bpmatrix%7D%201%26a%26b%5C%5C0%261%26c%5C%5C0%260%261%20%5Cend%7Bpmatrix%7D%5Ep%24%20%24%3D%5Cbegin%7Bpmatrix%7D%201%26pa%26pb%2B%5Csum%5Climits_%7Bi%3D1%7D%5E%7Bp-1%7Di%5Ccdot%20ac%20%5C%5C%200%261%26pc%20%5C%5C%200%260%261%20%5Cend%7Bpmatrix%7D%24%20which%20equals%20the%20identity%20matrix%20already%20for%20any%20%24a%2Cb%2Cc%24%2C%20because%20the%20coefficient%20are%20all%20divisible%20by%20%24p%24%20(%24%5Csum%5Climits_%7Bi%3D1%7D%5E%7Bp-1%7Di%24%20%24%3D%201%2B2%2B%5Cdots%2Bp-1%24%2C%20add%20first%20with%20last%2C%20second%20to%20second-last%2C%20etc%2C%20since%20%24p%24%20is%20odd%2C%20this%20equals%20%24p%2Bp%2B%5Cdots%2Bp%24%20is%20divisible%20by%20%24p%24).%20Thus%20any%20matrix%20in%20%24U%24%20has%20at%20most%20order%20%24p%24%2C%20thus%20cannot%20have%20order%20%24p%5E2%24.%0A%0A%20%20%20%20If%20%24p%20%3D%202%24%2C%20we%20may%20identify%20%24%24%5Cbegin%7Bpmatrix%7D%201%260%260%20%5C%5C%200%261%260%20%5C%5C%200%260%261%20%5Cend%7Bpmatrix%7D%20%3D%201%2C%5Cbegin%7Bpmatrix%7D%201%261%260%20%5C%5C%200%261%260%20%5C%5C%200%260%261%20%5Cend%7Bpmatrix%7D%20%3D%20a%2C%5Cbegin%7Bpmatrix%7D%201%260%261%20%5C%5C%200%261%260%20%5C%5C%200%260%261%20%5Cend%7Bpmatrix%7D%20%3D%20b%24%24%20and%20verify%20that%20%24a%5E4%20%3D%20b%5E2%20%3D%201%24%2C%20%24b%5E%7B-1%7Dab%20%3D%20a%5E%7B-1%7D%24%2C%20and%20that%20this%20group%20is%20isomorphic%20to%20the%20dihedral%20group%20%24D_4%24.%0A%0A%20%20%20%20%23%23%20P174.9%0A%0A%20%20%20%20**Let%20%24G%20%3D%20GL(2%2C%20q)%24%20where%20%24q%24%20is%20any%20prime%20power.%20Then%20%24%7C%20G%20%7C%20%3D%20q(q-1)%5E2(q%2B1)%24.%20Denote%20by%20%24m_g(X)%24%20the%20minimal%20polynomial%20of%20an%20element%20%24g%20%5Cin%20G%24.%20Show%20that%20if%20%24m_g(X)%20%3D%20X-a%24%20for%20some%20%24a%20%5Cin%20%5Cmathbb%7BF%7D_q%5E%7B%5Ctimes%7D%24%2C%20then%20%24g%20%3D%20%5Cbegin%7Bpmatrix%7D%20a%260%5C%5C0%26a%20%5Cend%7Bpmatrix%7D%20%5Cin%20Z(G)%24**.%0A%0A%20%20%20%20Proof%0A%0A%20%20%20%20We%20will%20be%20doing%20several%20matrices%20multiplications%20and%20solve%20several%20systems.%20For%20convenience%2C%20say%20%24g%20%3D%20%5Cbegin%7Bpmatrix%7D%20x%26y%5C%5Cz%26w%20%5Cend%7Bpmatrix%7D%24%20in%20matrix%20form.%0A%0A%20%20%20%20By%20assumption%20we%20have%20%24g%20-%20aI%20%3D%200%24%2C%20i.e.%20%24%24%5Cbegin%7Bpmatrix%7D%20x%26y%5C%5Cz%26w%20%5Cend%7Bpmatrix%7D-%5Cbegin%7Bpmatrix%7D%20a%260%5C%5C0%26a%20%5Cend%7Bpmatrix%7D%20%3D%20%5Cbegin%7Bpmatrix%7D%200%260%5C%5C0%260%20%5Cend%7Bpmatrix%7D%24%24%20so%20%24x%20%3D%20a%24%2C%20%24y%20%3D%200%24%2C%20%24z%20%3D%200%24%2C%20and%20%24w%20%3D%20a%24%2C%20thus%20%24g%24%20is%20of%20the%20form%20%24%5Cbegin%7Bpmatrix%7D%20a%260%5C%5C0%26a%20%5Cend%7Bpmatrix%7D%24.%20Since%20%24g%24%20of%20this%20form%20can%20be%20written%20as%20%24aI%24%20with%20inverse%20%24%5Cfrac%7B1%7D%7Ba%7DI%24%2C%20it%20must%20lie%20in%20the%20center.%20In%20particular%2C%20those%20%24g%24's%20are%20in%20their%20own%20conjugacy%20classes%2C%20and%20there%20are%20%24q-1%24%20of%20them%2C%20one%20for%20each%20%24a%24.%0A%0A%20%20%20%20%23%23%20P174.10%0A%0A%20%20%20%20**Show%20that%20if%20%24m_g(X)%20%3D%20(X-a)%5E2%24%20for%20some%20%24a%20%5Cin%20%5Cmathbb%7BF%7D_q%5E%7B%5Ctimes%7D%24%2C%20then%20%24g%24%20is%20conjugate%20with%20%24%5Cbegin%7Bpmatrix%7D%20a%261%5C%5C0%26a%20%5Cend%7Bpmatrix%7D%24%2C%20and%20the%20conjugacy%20class%20of%20%24g%24%20has%20order%20%24q%5E2%20-%201%24**.%0A%0A%20%20%20%20Proof%0A%0A%20%20%20%20Expand%20%24(g-aI)%5E2%20%3D%200%24%2C%20we%20will%20get%20a%20system%20%24%24%5Cbegin%7Bcases%7D%20(x-a)%5E2%2Byz%20%26%3D%200%5C%5C%20y(w%2Bx-2a)%20%26%3D%200%5C%5Cz(w%2Bx-2a)%20%26%3D0%5C%5C%20(w-a)%5E2%2Byz%20%26%3D%200%5Cend%7Bcases%7D.%24%24%20Since%20%24m_g(X)%24%20is%20assumed%20not%20%24X-a%24%2C%20we%20cannot%20have%20both%20%24y%2C%20z%20%3D%200%24.%20If%20they%20are%20both%20not%20zero%2C%20then%20%24x%3Dw%20%3D%20%5Csqrt%7B-yz%7D%2Ba%20%5Cne%20a%24%20but%20%24x%2Bw%20%3D%202a%24%2C%20which%20is%20impossible.%20So%20one%20of%20them%20is%20zero%20and%20the%20other%20is%20not%2C%20so%20%24g%24%20is%20of%20the%20form%20%24%24g%20%3D%20%5Cbegin%7Bpmatrix%7D%20a%26b%5C%5C0%26a%20%5Cend%7Bpmatrix%7D%5Ctext%7B%20or%20%7D%5Cbegin%7Bpmatrix%7D%20a%260%5C%5Cb%26a%20%5Cend%7Bpmatrix%7D%2C%20b%20%5Cne%200.%24%24%20It%20is%20not%20hard%20to%20verify%20they%20are%20both%20conjugate%20to%20%24%5Cbegin%7Bpmatrix%7D%20a%261%5C%5C0%26a%20%5Cend%7Bpmatrix%7D%24%2C%20for%20example%3A%20%24%24%5Cbegin%7Bpmatrix%7D%20b%260%5C%5C0%261%20%5Cend%7Bpmatrix%7D%5E%7B-1%7D%5Cbegin%7Bpmatrix%7D%20a%26b%5C%5C0%26a%20%5Cend%7Bpmatrix%7D%5Cbegin%7Bpmatrix%7D%20b%260%5C%5C0%261%20%5Cend%7Bpmatrix%7D%20%3D%20%5Cbegin%7Bpmatrix%7D%20a%261%5C%5C0%26a%20%5Cend%7Bpmatrix%7D.%24%24%20To%20see%20the%20order%20of%20the%20conjugacy%20class%2C%20by%20direct%20calculation%20it%20is%20not%20hard%20to%20see%20the%20stabilizer%20of%20%24%5Cbegin%7Bpmatrix%7D%20a%261%5C%5C0%26a%20%5Cend%7Bpmatrix%7D%24%20is%20of%20the%20form%20%24%5Cbegin%7Bpmatrix%7D%20x%26y%5C%5C0%26x%20%5Cend%7Bpmatrix%7D%24%3B%20%24x%24%20can%20be%20taken%20in%20%24%5Cmathbb%7BF%7D_q%5E%7B%5Ctimes%7D%24%20and%20%24y%24%20is%20completely%20arbitrary%2C%20so%20the%20order%20of%20the%20stabilizer%20is%20%24q(q-1)%24%2C%20thus%20orbit-stabilizer%20theorem%20tells%20us%20the%20order%20of%20the%20conjugacy%20class%20is%20%24(q-1)(q%2B1)%20%3D%20q%5E2-1%24.%0A%0A%20%20%20%20There%20are%20%24q-1%24%20of%20such%20conjugacy%20classes%2C%20one%20for%20each%20%24a%24.%0A%0A%20%20%20%20%23%23%20P174.11%0A%0A%20%20%20%20**Show%20that%20if%20%24m_g(X)%20%3D%20(X-a)(X-b)%24%20for%20some%20distinct%20%24a%2Cb%20%5Cin%20%5Cmathbb%7BF%7D_q%5E%7B%5Ctimes%7D%24%20then%20%24g%24%20is%20conjugate%20with%20%24%5Cbegin%7Bpmatrix%7D%20a%260%5C%5C0%26b%20%5Cend%7Bpmatrix%7D%24%20and%20the%20conjugacy%20class%20of%20%24g%24%20has%20order%20%24q%5E2%2Bq%24**.%0A%0A%20%20%20%20Proof%0A%0A%20%20%20%20Similarly%20we%20get%20a%20system%20%24%24%5Cbegin%7Bcases%7Dab%2Bx%5E2-ax-bx%2Byz%20%26%3D%200%20%5C%5C%20y(x%2Bw-a-b)%20%26%3D%200%20%5C%5C%20z(x%2Bw-a-b)%20%26%3D%200%20%5C%5C%20ab%2Bw%5E2-aw-bw%2Byz%20%26%3D%200%5Cend%7Bcases%7D.%24%24%20Suppose%20%24y%2Cz%24%20are%20both%20non-zero%2C%20then%20%24x%2Cw%20%3D%20%5Cfrac%7Ba%2Bb%5Cpm%5Csqrt%7B%5Cdots%7D%7D%7B2%7D%24%20by%20quadratic%20formula%2C%20but%20a%20matrix%20of%20this%20form%20is%20conjugate%20to%20%24%5Cbegin%7Bpmatrix%7D%20%5Cfrac%7Ba%2Bb%7D%7B2%7D%261%20%5C%5C%200%26%5Cfrac%7Ba%2Bb%7D%7B2%7D%20%5Cend%7Bpmatrix%7D%24%20and%20falls%20in%20the%20above%20category.%20Otherwise%20at%20least%20one%20of%20%24y%2Cz%24%20is%20zero%2C%20forcing%20%24x%20%3D%20a%2C%20w%20%3Db%24%20or%20%24w%20%3Da%2C%20x%3D%20b%24.%20A%20direct%20calculation%20shows%20that%20matrices%20of%20the%20form%20%24%24%5Cbegin%7Bpmatrix%7D%20a%260%5C%5C0%26b%20%5Cend%7Bpmatrix%7D%2C%5Cbegin%7Bpmatrix%7D%20b%260%5C%5C0%26a%20%5Cend%7Bpmatrix%7D%2C%5Cbegin%7Bpmatrix%7D%20a%26c%5C%5C0%26b%20%5Cend%7Bpmatrix%7D%2C%5Cbegin%7Bpmatrix%7D%20a%260%5C%5Cc%26b%20%5Cend%7Bpmatrix%7D%24%24%20are%20all%20in%20the%20same%20conjugacy%20class%2C%20as%20desired.%0A%0A%20%20%20%20The%20order%20of%20the%20conjugacy%20class%20is%20calculated%20in%20a%20similar%20manner%20as%20well.%20We%20may%20calculate%20%24%24%5Cbegin%7Bpmatrix%7D%20x%26y%5C%5Cz%26w%20%5Cend%7Bpmatrix%7D%5E%7B-1%7D%5Cbegin%7Bpmatrix%7D%20a%260%5C%5C0%26b%20%5Cend%7Bpmatrix%7D%5Cbegin%7Bpmatrix%7D%20x%26y%5C%5Cz%26w%20%5Cend%7Bpmatrix%7D%20%3D%20%5Cbegin%7Bpmatrix%7D%20%5Cfrac%7Bawx-byz%7D%7Bwx-yz%7D%26%5Cfrac%7Bawy-bwy%7D%7Bwx-yz%7D%5C%5C%5Cfrac%7B-axz%2Bbxz%7D%7Bwx-yz%7D%26%5Cfrac%7Bbwx-ayz%7D%7Bwx-yz%7D%20%5Cend%7Bpmatrix%7D%2C%24%24%20we%20can%20see%20the%20stabilizer%20consists%20of%20matrix%20of%20the%20form%20%24%5Cbegin%7Bpmatrix%7D%20x%260%5C%5C0%26w%20%5Cend%7Bpmatrix%7D%24%20where%20%24x%2Cw%20%5Cin%20%5Cmathbb%7BF%7D_q%5E%7B%5Ctimes%7D%24%20so%20has%20order%20%24(q-1)%5E2%24%2C%20and%20thus%20the%20conjugacy%20class%20has%20order%20%24q(q%2B1)%20%3D%20q%5E2%2Bq%24.%0A%0A%20%20%20%20There%20are%20%24(q-1)(q-2)%2F2%24%20of%20distinct%20unordered%20such%20%24a%2Cb%24%2C%20so%20there%20are%20%24(q-1)(q-2)%24%20such%20conjugacy%20classes.%0A%0A%20%20%20%20%23%23%20P174.12%0A%0A%20%20%20%20**Show%20that%20if%20%24r%2Cs%20%5Cin%20%5Cmathbb%7BF%7D_q%24%20are%20such%20that%20%24X%5E2-rX-s%24%20is%20irreducible%2C%20then%20the%20conjugacy%20class%20of%20%24%5Cbegin%7Bpmatrix%7D%200%261%5C%5Cs%26r%20%5Cend%7Bpmatrix%7D%24%20has%20order%20%24q%5E2-q%24**.%0A%0A%20%20%20%20Proof%0A%0A%20%20%20%20Similarly%20we%20have%20a%20system%20%24%24%5Cbegin%7Bcases%7Dx%5E2-rx%2Byz-s%20%26%3D%200%20%5C%5C%20y(x%2Bw-r)%20%26%3D%200%20%5C%5C%20z(x%2Bw-r)%20%26%3D%200%20%5C%5C%20w%5E2-rw%2Byz-s%20%26%3D%200%5Cend%7Bcases%7D.%24%24%20It%20is%20impossible%20to%20have%20at%20least%20one%20of%20%24y%2Cz%24%20being%20zero%2C%20because%20that%20forces%20%24x%5E2-rx-s%20%3D0%24%2C%20which%20is%20an%20irreducible%20quadratic%20polynomial%2C%20implies%20that%20%24x%20%5Cnotin%20%5Cmathbb%7BF%7D_%7Bq%7D%24.%20So%20neither%20of%20%24y%2Cz%24%20is%20zero.%20Now%20if%20%24yz%20%5Cne%20s%24%2C%20or%20equivalently%20if%20both%20%24x%2Cw%24%20are%20not%20zero%2C%20then%20this%20matrix%20is%20conjugate%20to%20a%20certain%20matrix%20in%20category%20%242%24%3B%20so%20%24yz%3Ds%24%20and%20one%20of%20%24x%2Cw%24%20is%20%24r%24%2C%20the%20other%20is%20%240%24.%20It%20is%20not%20hard%20to%20verify%20%24%24%5Cbegin%7Bpmatrix%7D%200%261%5C%5Cs%26r%20%5Cend%7Bpmatrix%7D%2C%5Cbegin%7Bpmatrix%7D%20r%261%5C%5Cs%260%20%5Cend%7Bpmatrix%7D%2C%5Cbegin%7Bpmatrix%7D%200%26a%5C%5Cs%2Fa%26r%20%5Cend%7Bpmatrix%7D%2C%5Cbegin%7Bpmatrix%7D%20r%26a%5C%5Cs%2Fa%260%20%5Cend%7Bpmatrix%7D%24%24%20are%20all%20conjugate%2C%20as%20desired.%0A%0A%20%20%20%20Again%20we%20count%20the%20stabilizer%2C%20by%20considering%20%24%24%5Cbegin%7Baligned%7D%5Cbegin%7Bpmatrix%7D%20x%26y%5C%5Cz%26w%20%5Cend%7Bpmatrix%7D%5Cbegin%7Bpmatrix%7D%200%261%5C%5Cs%26r%20%5Cend%7Bpmatrix%7D%3D%5Cbegin%7Bpmatrix%7D%20sy%26x%2Bry%5C%5Csw%26rw%2Bz%20%5Cend%7Bpmatrix%7D%5C%5C%3D%5Cbegin%7Bpmatrix%7D%20z%26w%5C%5Csx%2Brz%26rw%2Bsy%20%5Cend%7Bpmatrix%7D%3D%5Cbegin%7Bpmatrix%7D%200%261%5C%5Cs%26r%20%5Cend%7Bpmatrix%7D%5Cbegin%7Bpmatrix%7D%20x%26y%5C%5Cz%26w%20%5Cend%7Bpmatrix%7D%5Cend%7Baligned%7D%2C%24%24%20we%20either%20have%20%24x%3Dw%2C%20y%3Dz%20%3D%200%24%20(there%20are%20%24q-1%24%20of%20such)%20or%20%24y%20%3Dz%2Fs%20%3D%20(w-x)%2Fr%24%2C%20where%20if%20%24y%20%5Cin%20%5Cmathbb%7BF%7D_q%5E%7B%5Ctimes%7D%24%20and%20%24w%20%5Cin%20%5Cmathbb%7BF%7D_q%24%20are%20chosen%2C%20%24x%2Cz%24%20are%20fixed.%20In%20total%20the%20order%20of%20stabilizer%20is%20%24(q-1)(q%2B1)%24%20thus%20order%20of%20conjugacy%20class%20is%20%24q%5E2-q%24.%0A%0A%20%20%20%20The%20number%20of%20such%20conjugacy%20class%20is%20the%20same%20as%20the%20number%20of%20irreducible%20monic%20quadratic%20polynomials%20in%20%24%5Cmathbb%7BF%7D_q%24%2C%20which%20equals%20%24q%5E2%24%20(total%20number%20of%20monic%20polynomial)%20minus%20%24%5Cfrac%7Bq(q%2B1)%7D%7B2%7D%24%20(number%20of%20polynomials%20of%20the%20form%20%24(x-r)(x-s)%24%20or%20%24(x-r)%5E2%24%2C%20i.e.%20reducible%20ones)%2C%20which%20equals%20%24q(q-1)%2F2%24.%0A%0A%20%20%20%20%23%23%20P174.13%0A%0A%20%20%20%20**Verify%20that%20what%20follows%20is%20a%20complete%20list%20of%20the%20conjugacy%20classes%20of%20%24G%24%3A%20%24q-1%24%20classes%20of%20%241%24%20element%20each%2C%20indexed%20by%20elements%20of%20%24%5Cmathbb%7BF%7D_q%5E%7B%5Ctimes%7D%24%3B%20%24q-1%24%20classes%20of%20%24q%5E2-1%24%20elements%20each%2C%20indexed%20by%20elements%20of%20%24%5Cmathbb%7BF%7D_q%5E%7B%5Ctimes%7D%24%3B%20%24(q-1)(q-2)%2F2%24%20classes%20of%20%24q%5E2%2Bq%24%20elements%20each%2C%20indexed%20by%20unordered%20pairs%20of%20distinct%20elements%20of%20%24%5Cmathbb%7BF%7D_q%5E%7B%5Ctimes%7D%24%3B%20and%20%24q(q-1)%2F2%24%20classes%20of%20%24q%5E2-q%24%20elements%20each%2C%20indexed%20by%20irreducible%20monic%20quadratic%20polynomials%20with%20coefficients%20in%20%24%5Cmathbb%7BF%7D_q%24**.%0A%0A%20%20%20%20Proof%0A%0A%20%20%20%20This%20is%20an%20immediate%20result%20followed%20by%20preceding%20problems%2C%20and%20just%20to%20make%20sure%3A%20calculate%20%24%24%5Cbegin%7Baligned%7D(q-1)%5Ccdot%201%20%2B%20(q-1)%5Ccdot(q%5E2-1)%20%5C%5C%2B%20(q-1)(q-2)(q%5E2%2Bq)%2F2%20%2B%20q(q-1)%26(q%5E2-q)%2F2%20%5C%5C%26%3D%20q(q-1)%5E2(q%2B1)%5Cend%7Baligned%7D%24%24%20so%20indeed%20we%20have%20everything%20covered.%0A%20%20%20%20%22%22%22)%0A%20%20%20%20return%0A%0A%0A%40app.cell%0Adef%20_(mo)%3A%0A%20%20%20%20mo.md(r%22%22%22%0A%20%20%20%20%23%20Spring%20Semester%20Homework%202%0A%20%20%20%20%22%22%22)%0A%20%20%20%20return%0A%0A%0A%40app.cell%0Adef%20_(mo)%3A%0A%20%20%20%20mo.md(r%22%22%22%0A%20%20%20%20%23%23%20Exercise%201%0A%0A%20%20%20%20**Let%20%24G%20%3D%20%5Cmathbb%7BZ%7D_3%20%3D%20%5Clangle%20a%20%5Crangle%24%2C%20%24V%20%3D%20%5Cmathbb%7BC%7DG%24%20viewed%20as%20a%20%24%5Cmathbb%7BC%7DG%24-module.%20Check%20that%20%24a%24%20acts%20on%20%24V%24%20by**%20%24%24%5Calpha%20%3D%20%5Cbegin%7Bpmatrix%7D%200%260%261%5C%5C1%260%260%5C%5C0%261%260%20%5Cend%7Bpmatrix%7D.%24%24%20**Find%20three%20one-dimensional%20submodules%20of%20%24V%24%20and%20conclude%20%24V%24%20is%20semi-simple**.%0A%0A%20%20%20%20Solution%0A%0A%20%20%20%20View%20%24v%20%5Cin%20V%20%3D%20%5Cmathbb%7BC%7DG%24%20as%20vectors%20%24%5Clangle%20x%2Cy%2Cz%20%5Crangle%24%20then%20%241%5Ccdot%20v%20%3D%20%5Clangle%20x%2Cy%2Cz%20%5Crangle%24%3B%20since%20%24%5Calpha%24%20(thus%20%24%5Calpha%5E2%24%20and%20%24%5Calpha%5E3%20%3D%20I%24)%20can%20be%20viewed%20as%20permutation%2C%20%24G%24%20acts%20on%20%24V%24.%0A%0A%20%20%20%20If%20we%20take%20%24u_1%20%3D%201%20%2B%20a%20%2B%20a%5E2%24%20then%20certainly%20%24%5Cmathbb%7BC%7Du_1%24%20is%20a%20one-dimensional%20submodule.%20There%20are%20two%20others%20using%20the%20other%20two%20third%20roots%20of%20unity%3A%20if%20%24u_2%20%3D%201%2Be%5E%7B%5Cfrac%7B2%7D%7B3%7D%5Cpi%20i%7Da%20%2B%20e%5E%7B%5Cfrac%7B4%7D%7B3%7D%5Cpi%20i%7Da%5E2%24%20then%20%24au_2%20%3D%20e%5E%7B%5Cfrac%7B4%7D%7B3%7D%5Cpi%20i%7D%20%2B%20a%20%2B%20e%5E%7B%5Cfrac%7B2%7D%7B3%7D%5Cpi%20i%7Da%5E2%24%20%24%3D%20e%5E%7B%5Cfrac%7B4%7D%7B3%7D%5Cpi%20i%7Du_2%24%2C%20because%20%24e%5E%7B%5Cfrac%7B6%7D%7B3%7D%5Cpi%20i%7D%20%3D%201%24%3B%20similarly%20for%20%24u_3%20%3D%201%20%2B%20e%5E%7B-%5Cfrac%7B2%7D%7B3%7D%5Cpi%20i%7Da%20%2B%20e%5E%7B-%5Cfrac%7B4%7D%7B3%7D%5Cpi%20i%7Da%5E2%24.%20It%20is%20easy%20to%20see%20%24u_1%2C%20u_2%2C%20u_3%24%20are%20linearly%20independent%3B%20since%20the%20dimension%20matches%20we%20have%20%24V%20%3D%20%5Cmathbb%7BC%7Du_1%20%5Coplus%20%5Cmathbb%7BC%7Du_2%20%5Coplus%20%5Cmathbb%7BC%7Du_3%24%2C%20so%20%24V%24%20is%20a%20(direct)%20sum%20of%20simple%20modules%20(one-dimensional%20submodules%20is%20always%20simple)%2C%20thus%20is%20semi-simple.%0A%0A%20%20%20%20%23%23%20Exercise%202%0A%0A%20%20%20%20**Let%20%24R%20%3D%20F%5BX%5D%24%20and%20define%20three%20different%20%24R%24-module%20structures%20on%20%24F%5E2%24%20(call%20them%20%24V_1%2C%20V_2%2C%20V_3%24)%2C%20with%20%24X%24%20acting%20by**%20%24%24%5Calpha_1%20%3D%20%5Cbegin%7Bpmatrix%7D%201%261%5C%5C0%261%20%5Cend%7Bpmatrix%7D%2C%20%5Calpha_2%20%3D%20%5Cbegin%7Bpmatrix%7D%200%261%5C%5C-1%260%20%5Cend%7Bpmatrix%7D%2C%20%5Calpha_3%20%3D%20%5Cbegin%7Bpmatrix%7D%200%261%5C%5C1%260%20%5Cend%7Bpmatrix%7D%24%24%20**respectively.%20Which%20of%20these%20modules%20are%20semi-simple%20and%20how%20do%20they%20decompose%20for**%3A%0A%0A%20%20%20%201.%20**%24F%20%3D%20%5Cmathbb%7BC%7D%24**%3F%0A%20%20%20%202.%20**%24F%20%3D%20%5Cmathbb%7BR%7D%24**%3F%0A%20%20%20%203.%20**%24F%20%3D%20%5Cmathbb%7BF%7D_5%24**%3F%0A%0A%20%20%20%20Solution%0A%0A%20%20%20%201.%20If%20%24F%20%3D%20%5Cmathbb%7BC%7D%24%3A%0A%20%20%20%20%091.%20%24e_1%24%20is%20an%20eigenvector%20for%20(any%20polynomial%20of)%20%24%5Calpha_1%24%2C%20so%20%24Fe_1%24%20is%20a%20submodule%20of%20%24F%5E2%24.%20Any%20vector%20%24%5Cbegin%7Bpmatrix%7D%20a%5C%5Cb%20%5Cend%7Bpmatrix%7D%2C%20b%20%5Cne%200%24%20cannot%20be%20an%20eigenvector%2C%20in%20particular%20there%20is%20no%20other%20non-trivial%20submodules%2C%20and%20thus%20%24V_1%24%20is%20not%20semi-simple%3B%0A%20%20%20%20%092.%20For%20%24%5Calpha_2%24%2C%20solve%20for%20the%20characteristic%20function%20we%20get%20%24%5Clambda%5E2%2B1%5E2%20%3D%200%24%20%24%5Cimplies%20%5Clambda%20%3D%20%5Cpm%20i%24%2C%20so%20that%20%24u_1%20%3D%20%5Cbegin%7Bpmatrix%7D%201%5C%5C%20i%20%5Cend%7Bpmatrix%7D%24%20and%20%24u_2%3D%5Cbegin%7Bpmatrix%7D%201%5C%5C-i%20%5Cend%7Bpmatrix%7D%24%20are%20always%20eigenvectors.%20They%20are%20independent%2C%20and%20we%20get%20two%20(simple)%20submodules%2C%20%24V_2%24%20is%20semi-simple%3B%0A%20%20%20%20%093.%20Similarly%2C%20%24%5Calpha_3%24%20always%20have%20%24u_1%20%3D%20%5Cbegin%7Bpmatrix%7D%201%5C%5C1%20%5Cend%7Bpmatrix%7D%24%2C%20%24u_2%20%3D%20%5Cbegin%7Bpmatrix%7D%201%5C%5C-1%20%5Cend%7Bpmatrix%7D%24%20as%20eigenvectors%2C%20so%20%24V_3%24%20is%20semi-simple%3B%0A%20%20%20%202.%20If%20%24F%20%3D%20%5Cmathbb%7BR%7D%24%3A%0A%20%20%20%20%091.%20The%20module%20given%20by%20%24%5Calpha_1%24%20is%20still%20not%20semi-simple%20for%20the%20same%20reason%3B%0A%20%20%20%20%092.%20For%20%24%5Calpha_2%24%20there%20is%20no%20solution%20for%20%24%5Clambda%5E2%20%2B%201%5E2%20%3D%200%24%20in%20%24%5Cmathbb%7BR%7D%24%2C%20in%20fact%2C%20%24V_2%24%20in%20this%20case%20is%20isomorphic%20to%20%24%5Cmathbb%7BR%7D%5Bx%5D%2F(x%5E2%2B1)%24.%20Since%20%24x%5E2%20%2B%201%24%20is%20irreducible%20thus%20the%20ideal%20is%20maximal%2C%20we%20have%20%24V_2%24%20is%20simple%20thus%20semi-simple%3B%0A%20%20%20%20%093.%20The%20module%20given%20by%20%24%5Calpha_3%24%20is%20still%20semi-simple%20for%20the%20same%20reason%3B%0A%20%20%20%203.%20If%20%24F%20%3D%20%5Cmathbb%7BF%7D_5%24%3A%0A%20%20%20%20%091.%20The%20module%20given%20by%20%24%5Calpha_1%24%20is%20still%20not%20semi-simple%3B%0A%20%20%20%20%092.%20For%20%24%5Calpha_2%24%2C%20we%20solve%20%24%5Clambda%5E2%20%2B%201%5E2%20%3D%200%24%20gives%20%24%5Clambda%20%3D%202%2C%203%24%2C%20correspond%20to%20%24u_1%20%3D%20%5Cbegin%7Bpmatrix%7D%203%5C%5C1%20%5Cend%7Bpmatrix%7D%24%20and%20%24u_2%20%3D%20%5Cbegin%7Bpmatrix%7D%202%5C%5C1%20%5Cend%7Bpmatrix%7D%24%20(for%20example%20%24%5Cbegin%7Bpmatrix%7D%200%261%5C%5C-1%260%20%5Cend%7Bpmatrix%7D%5Cbegin%7Bpmatrix%7D%203%5C%5C1%20%5Cend%7Bpmatrix%7D%20%3D%20%5Cbegin%7Bpmatrix%7D%201%5C%5C-3%20%5Cend%7Bpmatrix%7D%24%20%24%3D%5Cbegin%7Bpmatrix%7D%201%5C%5C2%20%5Cend%7Bpmatrix%7D%24%20%24%3D2%20%5Cbegin%7Bpmatrix%7D%203%5C%5C1%20%5Cend%7Bpmatrix%7D%24)%2C%20and%20that%20%24V_2%24%20is%20semi-simple%3B%0A%20%20%20%20%093.%20The%20module%20given%20by%20%24%5Calpha_3%24%20is%20still%20semi-simple.%0A%0A%20%20%20%20%23%23%20Exercise%203%0A%0A%20%20%20%20**Let%20%24G%20%3D%20D_8%20%3D%20%5Clangle%20a%2Cb%3A%20a%5E4%20%3D%20b%5E2%20%3D%201%2C%20b%5E%7B-1%7Dab%20%3D%20a%5E%7B-1%7D%20%5Crangle%24.%20Find%20three%20one-dimensional%20submodules%20%24%5Cmathbb%7BC%7Du_i%24%20of%20%24%5Cmathbb%7BC%7DG%24%20(%24i%20%3D%201%2C2%2C3%24)%20such%20that**%20%24%24%5Cbegin%7Baligned%7D%20au_1%20%26%3D%20u_1%2C%20%26bu_1%20%26%3D%20-u_1%20%5C%5C%20au_2%20%26%3D%20-u_2%2C%26bu_2%20%26%3D%20u_2%20%5C%5C%20au_3%20%26%3D%20-u_3%2C%20%26bu_3%20%26%3D%20-u_3%20%5Cend%7Baligned%7D.%24%24%0A%0A%20%20%20%20Solution%0A%0A%20%20%20%20Look%20at%20the%20Cayley%20table%20for%20%24D_8%24%2C%20we%20need%20two%20rows%20of%20it%3A%20%24%24%5Cbegin%7Barray%7D%20%7B%7Cr%7Cr%7Cr%7Cr%7Cr%7Cr%7Cr%7Cr%7Cr%7C%7D%5Chline%20%20%26%201%20%26%20a%20%26%20a%5E2%20%26%20a%5E3%20%26%20b%20%26%20ab%20%26%20a%5E2b%20%26%20a%5E3b%20%5C%5C%20%5Chline%20a%20%26%20a%20%26%20a%5E2%20%26%20a%5E3%20%26%201%20%26%20ab%20%26%20a%5E2b%20%26%20a%5E3b%20%26%20b%20%5C%5C%20%5Chline%20b%20%26%20b%20%26%20a%5E3b%20%26%20a%5E2b%20%26%20ab%20%26%201%20%26%20a%5E3%20%26%20a%5E2%20%26%20a%20%5C%5C%20%5Chline%20%5Cend%7Barray%7D%24%24%20Then%20for%20example%3A%20%24a1%20%3D%20a%24%2C%20thus%20if%20%24au_1%20%3D%20u_1%24%20then%20the%20coefficient%20of%20%241%24%20in%20%24u_1%24%20must%20be%20the%20same%20as%20the%20coefficient%20of%20%24a%24%20in%20%24u_1%24%2C%20and%20so%20on.%20We%20should%20be%20able%20to%20find%3A%0A%0A%20%20%20%201.%20If%20we%20take%20%24u_1%20%3D%201%2Ba%2Ba%5E2%2Ba%5E3-b-ab-a%5E2b-a%5E3b%24%2C%20then%20%24au_1%20%3D%20a%2Ba%5E2%2Ba%5E3%2B1-ab-a%5E2b-a%5E3b-b%20%3D%20u_1%24%2C%20and%20%24bu_1%20%3D%20b%2Ba%5E3b%2Ba%5E2b%2Bab-1-a%5E3-a%5E2-a%20%3D%20-u_1%24%3B%0A%20%20%20%202.%20Similarly%20if%20%24u_2%20%3D%201-a%2Ba%5E2-a%5E3%2Bb-ab%2Ba%5E2b-a%5E3b%24%2C%20then%20%24au_2%20%3D%20a-a%5E2%2Ba%5E3-1%2Bab-a%5E2b%2Ba%5E3b-b%20%3D%20-u_2%24%2C%20and%20%24bu_2%20%3D%20b-a%5E3b%2Ba%5E2b-ab%2B1-a%5E3%2Ba%5E2-a%20%3D%20u_2%24%3B%0A%20%20%20%203.%20And%20if%20%24u_3%20%3D%201-a%2Ba%5E2-a%5E3-b%2Bab-a%5E2b%2Ba%5E3b%24%2C%20then%20%24au_3%20%3D%20a-a%5E2%2Ba%5E3-1-ab%2Ba%5E2b-a%5E3b%2Bb%20%3D%20-u_3%24%2C%20and%20%24bu_3%20%3D%20b-a%5E3b%2Ba%5E2b-ab-1%2Ba%5E3-a%5E2%2Ba%20%3D%20-u_3%24.%0A%0A%20%20%20%20And%20then%20%24%5Cmathbb%7BC%7Du_i%24%20are%20the%20one-dimensional%20submodules%20we%20want.%0A%0A%20%20%20%20%23%23%20Exercise%204%20(P145.3)%0A%0A%20%20%20%20**Show%20that%20if%20%24S%24%20is%20a%20simple%20%24%5Cmathbb%7BC%7DG%24-module%20and%20%24U%24%20is%20a%20one-dimensional%20%24%5Cmathbb%7BC%7DG%24-module%2C%20then%20%24S%20%5Cotimes%20U%24%20is%20simple**.%0A%0A%20%20%20%20Proof%0A%0A%20%20%20%20Since%20%24U%24%20is%20one-dimensional%20(i.e.%20%24U%20%3D%20%5Cmathbb%7BC%7Du%24)%2C%20by%20definition%20we%20know%20%24S%20%5Cotimes%20U%20%3D%24%20%24span%5Clbrace%20s_i%20%5Cotimes%20u%20%7C%20i%20%5Cin%20I%20%5Crbrace%24%20for%20some%20index%20set%20%24I%24.%20%24G%24%20acts%20on%20%24S%20%5Cotimes%20U%24%20by%20%24g(s_i%20%5Cotimes%20u)%20%3D%20(gs_i)%20%5Cotimes%20(gu)%24.%20Suppose%20%24V%24%20is%20a%20submodule%20of%20%24S%20%5Cotimes%20U%24%2C%20then%20it%20must%20be%20%24span%5Clbrace%20s_j%20%5Cotimes%20u%20%7C%20j%20%5Cin%20J%20%5Crbrace%24%20for%20some%20index%20set%20%24J%20%5Csubset%20I%24%20(because%20it%20is%20a%20subgroup%20in%20the%20first%20place).%20That%20is%20to%20say%2C%20since%20%24G%24%20acts%20on%20%24V%24%2C%20we%20need%20%24g(s_j%20%5Cotimes%20u)%20%3D%20(gs_j)%20%5Cotimes%20(gu)%20%5Cin%20V%24%20to%20make%20%24V%24%20a%20submodule.%20So%20%24gs_j%20%5Cin%20span%5Clbrace%20s_j%20%7C%20j%20%5Cin%20J%20%5Crbrace%24%20for%20any%20%24g%24%2C%20which%20by%20definition%20means%20%24span%5Clbrace%20s_j%20%5Crbrace%24%20is%20a%20submodule%20of%20%24S%24.%0A%0A%20%20%20%20The%20upshot%20is%20that%20%24V%24%20is%20of%20the%20form%20%24S'%20%5Cotimes%20U%24%20where%20%24S'%24%20is%20a%20submodule%20of%20%24S%24.%20But%20%24S%24%20is%20by%20assumption%20simple%2C%20thus%20%24V%24%20is%20either%20%24S%20%5Cotimes%20U%24%20or%20%240%20%5Cotimes%20U%20%5Ccong%200%24%2C%20which%20means%20%24S%20%5Cotimes%20U%24%20has%20no%20non-trivial%20submodule%2C%20so%20by%20definition%20it%20is%20simple.%0A%0A%20%20%20%20Another%20method%20is%20to%20let%20%24%5Cchi_S%2C%20%5Cchi_U%24%20be%20the%20corresponding%20irreducible%20characters%20such%20that%20%24%5Cchi_U(1)%20%3D%201%24%2C%20and%20check%20%24%5Cchi_S%5Cchi_U%24%20is%20an%20irreducible%20character.%0A%0A%20%20%20%20%23%23%20Exercise%205%0A%0A%20%20%20%20**Let%20%24n%20%5Cge%203%24%20be%20odd%2C%20consider%20the%20group%20%24H%20%3D%20H_%7B8n%7D%24%20of%20order%20%248n%24%20given%20by**%20%24%24H_%7B8n%7D%3D%5Clangle%20a%2Cb%3Aa%5E%7B2n%7D%20%3D%20b%5E4%20%3D%201%2Cba%20%3D%20a%5E%7B-1%7Db%5E%7B-1%7D%2Cb%5E%7B-1%7Da%20%3D%20a%5E%7B-1%7Db%20%5Crangle.%24%24%0A%0A%20%20%20%201.%20**Show%20that%20if%20%24%5Cvarepsilon%24%20is%20any%20%24n%24-th%20root%20of%20unity%20in%20%24%5Cmathbb%7BC%7D%24%2C%20then**%20%24%24a%20%5Cto%20%5Cbegin%7Bpmatrix%7D%20%5Cvarepsilon%260%5C%5C0%26-%5Cvarepsilon%5E%7B-1%7D%20%5Cend%7Bpmatrix%7D%2C%20%20b%20%5Cto%20%5Cbegin%7Bpmatrix%7D%200%261%5C%5C-1%260%20%5Cend%7Bpmatrix%7D%24%24%20**is%20a%20representation%20of%20%24H%24.%20Check%20that%20all%20these%20representations%20are%20irreducible%20and%20non-isomorphic**%3B%0A%20%20%20%202.%20**There%20are%20further%20%24n-1%24%20irreducible%20representations%20of%20%24H%24%2C%20which%20are%20associated%20to%20the%20%242n%24-th%20roots%20of%20unity%20%24%5Ceta%20%3D%20e%5E%7B%5Cpi%20i%20s%2Fn%7D%2C%201%20%5Cle%20s%20%5Cle%20n-1%24.%20In%20these%20representations%20%24b%20%5Cto%20%5Cbegin%7Bpmatrix%7D%200%261%5C%5C1%260%20%5Cend%7Bpmatrix%7D%24.%20Write%20them%20down%20and%20explain%20why%20they%20are%20irreducible%2C%20distinct%20and%20different%20from%20the%20ones%20in%20part%201**%3B%0A%20%20%20%203.%20**Now%20you%20almost%20have%20complete%20list%20of%20simple%20%24%5Cmathbb%7BC%7DH%24-modules%2C%20but%20not%20quite.%20Complete%20it**.%0A%0A%20%20%20%20Proof%0A%0A%20%20%20%201.%20Let's%20denote%20%24A%2CB%24%20as%20the%20two%20matrices%20and%20do%20some%20matrix%20multiplication%3A%20%24A%5Em%20%3D%20%5Cbegin%7Bpmatrix%7D%20%5Cvarepsilon%5Em%260%5C%5C0%26(-1)%5Em%5Cvarepsilon%5E%7B-m%7D%20%5Cend%7Bpmatrix%7D%24.%20Since%20%24n%24%20is%20odd%2C%20we%20must%20have%20%24A%5E%7B2n%7D%20%3D%20I%24%20and%20no%20smaller%20%24m%24%20gives%20%24A%5Em%20%3D%20I%24.%20Also%20%24B%5E4%20%3D%20I%24%20and%20%24B%5E2%2C%20B%5E3%20%5Cne%20I%24.%20So%20the%20orders%20match.%20Also%20%24BA%20%3D%20%5Cbegin%7Bpmatrix%7D%200%26-%5Cvarepsilon%5E%7B-1%7D%5C%5C-%5Cvarepsilon%260%20%5Cend%7Bpmatrix%7D%20%3D%20A%5E%7B-1%7DB%5E%7B-1%7D%24%2C%20and%20%24B%5E%7B-1%7DA%20%3D%20%5Cbegin%7Bpmatrix%7D%200%26%5Cvarepsilon%5E%7B-1%7D%5C%5C%5Cvarepsilon%260%20%5Cend%7Bpmatrix%7D%20%3DA%5E%7B-1%7DB%24.%20So%20the%20relations%20also%20match%2C%20thus%20we%20have%20a%20representation.%0A%20%20%20%20%09If%20%24A%2C%20B%24%20are%20reducible%20then%20they%20are%20simultaneously%20diagonalizable%2C%20which%20happens%20if%20and%20only%20if%20they%20commute%2C%20and%20which%20implies%20they%20share%20an%20eigenvector.%20However%20%24A%24%20has%20eigenvector%20%24%5Cbegin%7Bpmatrix%7D%201%5C%5C0%20%5Cend%7Bpmatrix%7D%2C%20%5Cbegin%7Bpmatrix%7D%200%5C%5C1%20%5Cend%7Bpmatrix%7D%24%20while%20%24B%24%20has%20%24%5Cbegin%7Bpmatrix%7D1%5C%5Ci%20%20%5Cend%7Bpmatrix%7D%2C%5Cbegin%7Bpmatrix%7D%20%20-1%5C%5Ci%5Cend%7Bpmatrix%7D%24.%20So%20this%20is%20an%20irreducible%20representation.%0A%20%20%20%20%09Isomorphic%20representations%20have%20the%20same%20character.%20Since%20%24%5Cvarepsilon%20-%20%5Cvarepsilon%5E%7B-1%7D%24%20%24%3D%20e%5E%7B%5Cpi%20i%202%20m%2Fn%7D%20-%20e%5E%7B-%5Cpi%20i%202%20m%2Fn%7D%24%20%24%3D2i%20%5Csin(%5Cfrac%7B2%5Cpi%20m%7D%7Bn%7D)%24%20and%20if%20%24n%5Cge%203%24%20is%20odd%2C%20those%20values%20are%20always%20distinct%20if%20%242m%2Fn%20%5Cle%202%24%20(because%20they%20need%20to%20be%20symmetric%20about%20%24%5Cfrac%7B1%7D%7B2%7D%24%20or%20%24%5Cfrac%7B3%7D%7B2%7D%24%2C%20which%20never%20happen%20if%20%24n%24%20is%20odd)%2C%20thus%20none%20of%20those%20representations%20are%20isomorphic%3B%0A%20%20%20%202.%20Those%20representations%20map%20%24a%24%20to%20%24A%20%3D%20%5Cbegin%7Bpmatrix%7D%20%5Ceta%260%5C%5C0%26%5Ceta%5E%7B-1%7D%20%5Cend%7Bpmatrix%7D%24.%20Since%20%24A%5Em%20%3D%20%5Cbegin%7Bpmatrix%7D%20%5Ceta%5E%7Bm%7D%260%5C%5C0%26%5Ceta%5E%7B-m%7D%20%5Cend%7Bpmatrix%7D%24%2C%20%24A%24%20has%20order%20%242n%24.%20%24B%24%20has%20order%20%242%20%7C%204%24%2C%20and%20the%20relations%20all%20match%2C%20so%20these%20are%20representations.%0A%20%20%20%20%09They%20are%20irreducible%20for%20the%20very%20same%20reason%3A%20these%20%24A%24's%20have%20eigenvectors%20%24%5Cbegin%7Bpmatrix%7D%201%5C%5C0%20%5Cend%7Bpmatrix%7D%2C%20%5Cbegin%7Bpmatrix%7D%200%5C%5C1%20%5Cend%7Bpmatrix%7D%24%20but%20%24B%24%20has%20%24%5Cbegin%7Bpmatrix%7D%201%5C%5C1%20%5Cend%7Bpmatrix%7D%2C%5Cbegin%7Bpmatrix%7D%201%5C%5C-1%20%5Cend%7Bpmatrix%7D%24.%20They%20are%20distinct%20again%20because%20the%20character%20is%20never%20the%20same%3A%20%24%5Ceta%2B%5Ceta%5E%7B-1%7D%24%20%24%3D2%5Ccos(%5Cpi%20m%2Fn)%24%20and%20cosine%20is%20strictly%20monotone%20on%20%24%5B0%2C%5Cpi%5D%24.%0A%20%20%20%20%09It%20also%20follows%20that%20they%20are%20not%20isomorphic%20to%20the%20representations%20in%20part%201%2C%20as%20%24%5Ceta%20%2B%20%5Ceta%5E%7B-1%7D%24%20is%20always%20real%2C%20but%20%24%5Cvarepsilon%20-%20%5Cvarepsilon%5E%7B-1%7D%24%20is%20always%20pure%20imaginary%20(they%20are%20possibly%20the%20same%20if%20they%20are%20both%20%240%24%2C%20but%20since%20%24n%24%20is%20odd%2C%20this%20is%20impossible)%3B%0A%20%20%20%203.%20The%20above%20representations%2C%20viewed%20as%20submodules%2C%20are%20all%20of%20dimension%20%242%24.%20We%20know%20%248n%24%20should%20be%20the%20sum%20of%20squares%20of%20the%20dimensions%20of%20submodules%2C%20so%20%24%248n%20%3D%20n(2%5E2)%20%2B%20(n-1)(2%5E2)%20%2B%20%5Cdots%2C%24%24%20so%20we%20are%20still%20missing%20%244%24%20dimensions.%20Since%20there%20is%20always%20trivial%20module%2C%20our%20only%20option%20is%20to%20have%20%244%20%3D%201%5E2%20%2B%201%5E2%20%2B%201%5E2%20%2B%201%5E2%24.%20In%20fact%20those%20are%20easy%3A%20we%20just%20map%20%24a%2C%20b%24%20to%20%24%5Cpm%201%2C%20%5Cpm%201%24%2C%20those%20all%20give%20a%20one-dimensional%20submodule.%20There%20are%20%244%24%20of%20them%2C%20so%20the%20dimensions%20match%2C%20and%20we%20are%20done.%0A%0A%20%20%20%20%23%23%20Exercise%206%0A%20%20%20%20**Let%20%24G%20%3D%20F_%7B55%7D%20%3D%20%5Clangle%20a%2Cb%20%3A%20a%5E%7B11%7D%20%3D%20b%5E5%20%3D1%2C%20b%5E%7B-1%7Dab%20%3D%20a%5E3%20%5Crangle%24**.%0A%0A%20%20%20%201.%20**Show%20that%20the%20derived%20subgroup%20of%20%24G%24%20is%20%24G'%20%3D%20%5Clangle%20a%20%5Crangle%24.%20Check%20that%20%24G%24%20has%20%247%24%20conjugacy%20classes%20of%20sizes%20%241%2C5%2C5%2C11%2C11%2C11%2C11%24%20respectively.%20What%20are%20their%20representatives**%3F%0A%20%20%20%202.%20**Since%20%24G%2FG'%20%5Ccong%20%5Cmathbb%7BZ%7D_5%24%2C%20%24G%24%20has%20five%20linear%20characters.%20The%20remaining%20two%20simple%20%24%5Cmathbb%7BC%7DG%24-modules%20have%20dimensions%20%5C_%5C_%5C_%20and%20%5C_%5C_%5C_%20.%20These%20considerations%20together%20with%20the%20orthogonality%20relations%20are%20sufficient%20to%20complete%20the%20entire%20character%20table%20of%20%24G%24.%20Write%20it%20down**.%0A%0A%20%20%20%20Proof%0A%0A%20%20%20%201.%20By%20definition%20of%20the%20commutator%20subgroup%2C%20%24G'%24%20must%20contain%20%24a%5E%7B-1%7Db%5E%7B-1%7Dab%20%3D%20a%5E%7B-1%7Da%5E3%20%3D%20a%5E2%24%2C%20since%20%24%5Cgcd%7B2%2C11%7D%20%3D%201%24%2C%20%24G'%24%20must%20contain%20%24%5Clangle%20a%20%5Crangle%24%2C%20in%20particular%20%24%7C%20G'%20%7C%20%5Cge%2011%24.%20Since%20%24G'%24%20is%20a%20subgroup%20of%20%24G%24%20it%20has%20order%20%24%5Cle%2055%24%20and%20is%20a%20divisor%20of%20%2455%24.%20The%20only%20possible%20such%20numbers%20are%20%2411%24%20or%20%2455%24.%20It%20cannot%20be%20%2455%24%20because%20that%20would%20imply%20%24G%24%20is%20abelian%20itself%2C%20which%20is%20not%2C%20so%20%24G'%24%20has%20%2411%24%20elements%20so%20it%20must%20be%20%24%5Clangle%20a%20%5Crangle%24.%0A%20%20%20%20%09%241%24%20forms%20its%20own%20conjugacy%20class.%20%24G'%24%20as%20a%20normal%20subgroup%20is%20a%20disjoint%20union%20of%20conjugacy%20classes%3A%20%241%24%20lies%20inside%20of%20it%2C%20there%20are%20another%20two%20classes%20in%20%24G'%24%2C%20%24%5Clbrace%20a%2C%20a%5E3%2C%20a%5E9%2C%20a%5E%7B27%7D%20%3D%20a%5E5%2C%20a%5E%7B15%7D%20%3D%20a%5E4%20%5Crbrace%24%2C%20and%20%24%5Clbrace%20a%5E2%2C%20a%5E6%2C%20a%5E%7B18%7D%20%3D%20a%5E7%2C%20a%5E%7B21%7D%20%3D%20a%5E%7B10%7D%2C%20a%5E%7B30%7D%20%3D%20a%5E8%20%5Crbrace%24.%0A%20%20%20%20%09There%20are%20%244%24%20conjugacy%20classes%20left%2C%20the%20representatives%20are%20%24ab%2C%20ab%5E2%2C%20ab%5E3%2C%20ab%5E4%24%20respectively%2C%20and%20each%20conjugacy%20class%20contains%20%24%5Clbrace%20b%5Ei%2C%20ab%5Ei%2C%20a%5E2b%5E1%2C%20%5Cdots%2C%20a%5E%7B10%7Db%5Ei%20%5Crbrace%24%20for%20each%20%24i%24.%20For%20example%2C%20%24ab%24%20is%20conjugate%20to%20%24b%24%20by%20%24a%5E7b%24%2C%20and%20%24ab%24%20is%20conjugate%20to%20%24a%5E2b%24%20by%20%24a%5E6b%24%2C%20thus%20all%20%24a%5Ej%20b%24%20is%20conjugate%20to%20%24ab%24%3B%20similar%20for%20other%20powers%20of%20%24b%24.%0A%20%20%20%20%09In%20fact%2C%20since%20all%20%24a%5Ei%20b%5Ej%24%20for%20a%20fixed%20%24j%24%20is%20in%20the%20same%20conjugacy%20class%2C%20we%20may%20choose%20to%20represent%20them%20using%20%24b%2C%20b%5E2%2C%20b%5E3%24%20and%20%24b%5E4%24%2C%20and%20we%20can%20choose%20%24a%5E%7B10%7D%20%3D%20a%5E%7B-1%7D%24%20to%20represent%20the%20third%20conjugacy%20class%2C%20making%20it%20easier%20to%20see%20the%20relations%20in%20character%20table%3B%0A%20%20%20%202.%20So%20we%20need%20two%20numbers%20such%20that%20%2455%20%3D%205%20%5Ccdot%201%5E2%20%2B%20%E2%80%A6%5E2%20%2B%20%E2%80%A6%5E2%24%2C%20%241%5E2%20%2B%207%5E2%24%20wouldn't%20work%20as%20that%20adds%20a%20linear%20character.%20So%20the%20other%20two%20modules%20both%20have%20dimension%20%245%24.%0A%20%20%20%20%09We%20start%20to%20write%20the%20character%20table.%20First%20we%20can%20safely%20get%20the%20following%3A%20we%20know%20what%20%24%5Cchi_%7Breg%7D%24%20is%2C%20we%20know%20what%20%24%5Cchi_1%24%20is%3B%20character%20of%20%241%2C%20a%24%20and%20%24a%5E%7B-1%7D%24%20would%20be%20trivial%20on%20the%20linear%20character%20because%20they%20are%20elements%20of%20the%20commutator%20subgroup.%20And%20character%20of%20powers%20of%20%24b%24's%20has%20to%20be%20powers%20of%20%24w%20%3D%20e%5E%7B2%5Cpi%20i%2F5%7D%24's%20to%20match%20the%20order.%20We%20also%20know%20%24%5Cchi_6(1)%20%3D%20%5Cchi_7(1)%20%3D%205%24.%20Also%20%24%5Cchi_%7B6%2C7%7D(b%5Ei)%24%20must%20be%20%240%24%2C%20because%20%241%5E2%2Bw%5E2%2B%5Cdots%20%2B%20(w%5E4)%5E2%20%3D%200%24%20already.%20Also%20since%20%24a%24%20and%20%24a%5E%7B-1%7D%24%20are%20in%20different%20conjugacy%20classes%2C%20those%20values%20are%20needed%20to%20be%20complex%20conjugate.%20So%20we%20have%3A%20%24%24%5Cbegin%7Barray%7D%20%7B%7Cr%7Cr%7Cr%7Cr%7Cr%7Cr%7Cr%7Cr%7C%7D%5Chline%20%20%26%201%20%26%20a%20%26%20a%5E%7B-1%7D%20%26%20b%20%26%20b%5E2%20%26%20b%5E3%20%26%20b%5E4%20%5C%5C%20%5Chline%20C_i%20%26%2055%20%26%2011%20%26%2011%20%26%205%20%26%205%265%265%20%20%5C%5C%20%5Chline%20%5Cchi_1%20%26%201%20%26%201%20%26%201%20%26%201%20%26%201%20%26%201%20%26%201%20%5C%5C%20%5Chline%20%5Cchi_2%20%26%201%20%26%201%20%26%201%20%26%20w%20%26%20w%5E2%20%26%20w%5E3%20%26%20w%5E4%20%5C%5C%20%5Chline%20%5Cchi_3%20%26%201%20%26%201%20%26%201%20%26%20w%5E2%20%26%20w%5E4%20%26%20w%20%26%20w%5E3%20%5C%5C%20%5Chline%20%5Cchi_4%20%26%201%20%26%201%20%26%201%20%26%20w%5E3%20%26%20w%20%26%20w%5E4%20%26%20w%5E2%20%5C%5C%20%5Chline%20%5Cchi_5%20%26%201%20%26%201%20%26%201%20%26%20w%5E4%20%26%20w%5E3%20%26%20w%5E2%20%26%20w%20%5C%5C%20%5Chline%205%5Cchi_6%20%26%205%20%26%20c%20%26%20%5Coverline%7Bc%7D%20%26%200%20%26%200%20%26%200%20%26%200%20%5C%5C%20%5Chline%205%5Cchi_7%20%26%205%20%26%20%5Coverline%7Bc%7D%20%26%20c%20%26%200%20%26%200%20%26%200%20%26%200%20%5C%5C%20%5Chline%20%5Cchi_%7Breg%7D%20%26%2055%20%26%200%20%26%200%20%26%200%20%26%200%20%26%200%20%26%200%20%5C%5C%20%5Chline%20%20%5Cend%7Barray%7D%24%24%20Write%20%24c%20%3D%20x%2B%20y%20i%24%2C%20then%20%24%5Cchi_%7Breg%7D(a)%20%3D%200%24%20gives%20that%20%245%2B5x%2B5yi%20%2B%205x%20-%205y%20i%20%3D%200%24%20thus%20%24x%20%3D%20-%5Cfrac%7B1%7D%7B2%7D%24.%20Orthogonality%20relation%20%24%5Clangle%20%5Cchi_6%2C%20%5Cchi_6%20%5Crangle%20%3D%201%24%20gives%20that%20%241%20%3D%20%5Cfrac%7B25%7D%7B55%7D%20%2B%202%5Ccdot%5Cfrac%7B(x%2Byi)(x-yi)%7D%7B11%7D%24%20so%20%243%20%3D%20x%5E2%2By%5E2%24%20and%20%24y%20%3D%20%5Cfrac%7B%5Csqrt%7B11%7D%7D%7B2%7D%24%20%2C%20this%20gives%20the%20complete%20character%20table%20%24%24%5Cbegin%7Barray%7D%20%7B%7Cr%7Cr%7Cr%7Cr%7Cr%7Cr%7Cr%7Cr%7C%7D%5Chline%20%20%26%201%20%26%20a%20%26%20a%5E%7B-1%7D%20%26%20b%20%26%20b%5E2%20%26%20b%5E3%20%26%20b%5E4%20%5C%5C%20%5Chline%20C_i%20%26%2055%20%26%2011%20%26%2011%20%26%205%20%26%205%265%265%20%20%5C%5C%20%5Chline%20%5Cchi_1%20%26%201%20%26%201%20%26%201%20%26%201%20%26%201%20%26%201%20%26%201%20%5C%5C%20%5Chline%20%5Cchi_2%20%26%201%20%26%201%20%26%201%20%26%20w%20%26%20w%5E2%20%26%20w%5E3%20%26%20w%5E4%20%5C%5C%20%5Chline%20%5Cchi_3%20%26%201%20%26%201%20%26%201%20%26%20w%5E2%20%26%20w%5E4%20%26%20w%20%26%20w%5E3%20%5C%5C%20%5Chline%20%5Cchi_4%20%26%201%20%26%201%20%26%201%20%26%20w%5E3%20%26%20w%20%26%20w%5E4%20%26%20w%5E2%20%5C%5C%20%5Chline%20%5Cchi_5%20%26%201%20%26%201%20%26%201%20%26%20w%5E4%20%26%20w%5E3%20%26%20w%5E2%20%26%20w%20%5C%5C%20%5Chline%205%5Cchi_6%20%26%205%20%26%20%5Cfrac%7B-1%2B%5Csqrt%7B-11%7D%7D%7B2%7D%20%26%20%5Cfrac%7B-1-%5Csqrt%7B-11%7D%7D%7B2%7D%20%26%200%20%26%200%20%26%200%20%26%200%20%5C%5C%20%5Chline%205%5Cchi_7%20%26%205%20%26%20%5Cfrac%7B-1-%5Csqrt%7B-11%7D%7D%7B2%7D%20%26%20%5Cfrac%7B-1%2B%5Csqrt%7B-11%7D%7D%7B2%7D%20%26%200%20%26%200%20%26%200%20%26%200%20%5C%5C%20%5Chline%20%5Cchi_%7Breg%7D%20%26%2055%20%26%200%20%26%200%20%26%200%20%26%200%20%26%200%20%26%200%20%5C%5C%20%5Chline%20%20%5Cend%7Barray%7D%24%24%0A%0A%20%20%20%20%23%23%20Exercise%207%0A%0A%20%20%20%20**Let%20%24%5Cchi%24%20be%20the%20character%20of%20the%20%24%5Cmathbb%7BC%7DG%24-module%20%24V%24%2C%20and%20denote%20by%20%24%5Cchi_S%24%2C%20%24%5Cchi_%7B%5CLambda%7D%24%20the%20characters%20of%20the%20%24%5Cmathbb%7BC%7DG%24-modules%20%24S%5E2V%24%20and%20%24%5CLambda%5E2V%24%20respectively**.%0A%0A%20%20%20%201.%20**Show%20that%20for%20%24g%20%5Cin%20G%24%2C**%20%24%24%5Cbegin%7Baligned%7D2%5Cchi_S(g)%20%26%3D%20%5Cchi(g)%5E2%20%2B%20%5Cchi(g%5E2)%2C%5C%5C%202%5Cchi_%7B%5CLambda%7D(g)%20%26%3D%20%5Cchi(g)%5E2%20-%20%5Cchi(g%5E2)%3B%5Cend%7Baligned%7D%24%24%0A%20%20%20%202.%20**Let%20now%20%24%7C%20G%20%7C%20%3D%2024%24%20and%20%24g_i%20(1%20%5Cle%20i%20%5Cle%207)%24%20be%20the%20representatives%20of%20seven%20conjugacy%20classes%20%24%5Cmathcal%7BK%7D_i%24%20in%20%24G%24%20of%20sizes%20%241%2C1%2C6%2C4%2C4%2C4%2C4%24%20respectively.%20Moreover%20%24g_i%5E2%24%20is%20conjugate%20to%20%24g_1%2C%20g_1%2C%20g_2%2C%20g_5%2C%20g_4%2C%20g_4%2C%20g_5%24%20respectively.%20Let%20%24%5Cchi%24%20be%20the%20character%20of%20%24G%24%2C%20where%20%24%5Cchi(g_i)%24%20is%20%242%2C%20-2%2C%200%2C%20-w%5E2%2C%20-w%2Cw%2Cw%5E2%24%20respectively%20(%24w%20%3D%20e%5E%7B2%5Cpi%20i%2F3%7D%24).%20Is%20%24%5Cchi%24%20irreducible%3F%20Find%20%24%5Cchi_S%24%20and%20%24%5Cchi_%7B%5CLambda%7D%24%20and%20check%20whether%20they%20are%20irreducible.%20Complete%20the%20character%20table%20of%20%24G%24**.%0A%0A%20%20%20%20Proof%0A%20%20%20%201.%20Take%20%24g%20%5Cin%20G%24%20and%20choose%20basis%20%24%5Clbrace%20e_1%2C%5Cdots%2C%20e_k%20%5Crbrace%24%20of%20%24V%24%20such%20that%20%24g%24%20acts%20diagonally%2C%20i.e.%20%24ge_i%20%3D%20%5Clambda_ie_i%24.%20Now%20we%20know%20%24%5Clbrace%20e_i%20%5Cotimes%20e_j%20%2B%20e_j%20%5Cotimes%20e_i%20%7C%201%20%5Cle%20i%20%5Cle%20j%20%5Cle%20k%5Crbrace%24%20form%20a%20basis%20for%20%24S%5E2V%24%2C%20and%20we%20have%20%24%24g(e_i%20%5Cotimes%20e_j%20%2B%20e_j%20%5Cotimes%20e_i)%20%3D%20%5Clambda_ie_i%20%5Cotimes%20%5Clambda_je_j%20%2B%20%5Clambda_je_j%20%5Cotimes%20%5Clambda_ie_i%20%3D%20%5Clambda_i%5Clambda_j(e_i%20%5Cotimes%20e_j%20%2B%20e_j%20%5Cotimes%20e_i)%2C%24%24%20so%20%24%5Cchi_S(g)%24%20is%20just%20the%20sum%20of%20%24%5Clambda_i%5Clambda_j%24%20over%20%241%20%5Cle%20i%20%5Cle%20j%20%5Cle%20k%24%2C%20i.e.%20%24%24%5Cchi_S(g)%20%3D%20%5Csum_%7B1%20%5Cle%20i%20%5Cle%20j%20%5Cle%20k%7D%5Clambda_i%20%5Clambda_j.%24%24%20Similarly%2C%20we%20have%20%24%24%5Cchi_%7B%5CLambda%7D(g)%20%3D%20%5Csum_%7B1%20%5Cle%20i%20%3C%20j%20%5Cle%20k%7D%5Clambda_i%5Clambda_j%2C%24%24%20notice%20there%20is%20a%20slight%20difference%20in%20the%20summand%20indices.%0A%20%20%20%20%09Now%20we%20verify%20the%20above%20equalities%3A%0A%20%20%20%20%09By%20the%20construction%20we%20have%20%24%5Cchi(g)%20%3D%20%5Csum%5Climits_i%20%5Clambda_i%24%2C%20thus%20%24%5Cchi(g)%5E2%24%20%24%3D%20(%5Csum%20%5Clambda_i)%5E2%24%20%24%3D%20%5Csum%5Climits_i%20%5Clambda_i%5E2%20%2B%202%5Csum%5Climits_%7B1%20%5Cle%20i%20%3C%20j%20%5Cle%20k%7D%5Clambda_i%5Clambda_j%24.%20Also%20since%20%24ge_i%20%3D%20%5Clambda_ie_i%24%2C%20we%20have%20%24g%5E2e_i%20%3D%20%5Clambda_i%5E2e_i%24%2C%20so%20%24%5Cchi(g%5E2)%20%3D%20%5Csum%5Climits_i%20%5Clambda_i%5E2%24.%20It%20is%20straight-forward%20now%20to%20see%20%24%242%5Cchi_%7B%5CLambda%7D(g)%20%3D%20%5Cchi(g)%5E2%20-%20%5Cchi(g%5E2).%24%24%20The%20other%20equality%20%24%242%5Cchi_S(g)%20%3D%20%5Cchi(g)%5E2%20%2B%20%5Cchi(g%5E2)%24%24%20also%20follows%2C%20by%20merely%20noticing%20%24%5Csum%5Climits_i%5Clambda_i%5E2%20%3D%20%5Csum_%7B1%5Cle%20i%20%3D%20j%20%5Cle%20k%7D%5Clambda_i%5Clambda_j%24%3B%0A%20%20%20%202.%20The%20conjugacy%20classes%20have%20sizes%20%241%2C1%2C6%2C4%2C4%2C4%2C4%24%2C%20so%20the%20centralizers%20have%20sizes%20%2424%2C%2024%2C%204%2C%206%2C%206%2C%206%2C%206%24.%20First%20we%20check%20the%20orthogonality%20relation%3A%20%24%24%5Clangle%20%5Cchi%2C%20%5Cchi%20%5Crangle%20%3D%20%5Cfrac%7B4%7D%7B24%7D%20%2B%20%5Cfrac%7B4%7D%7B24%7D%20%2B%20%5Cfrac%7B1%7D%7B6%7D%20%2B%5Cfrac%7B1%7D%7B6%7D%20%2B%20%5Cfrac%7B1%7D%7B6%7D%2B%20%5Cfrac%7B1%7D%7B6%7D%20%3D%201%24%24%20(%24w%5Coverline%7Bw%7D%24%20is%20%241%24)%2C%20so%20%24%5Cchi%24%20is%20irreducible.%20From%20the%20conjugation%20information%20we%20have%2C%20we%20get%20%24%24%5Cchi(g%5E2)%20%3D%20%5Cchi(g_1)%2C%20%5Cchi(g_1)%2C%20%5Cchi(g_2)%2C%20%5Cchi(g_5)%2C%20%5Cchi(g_4)%2C%20%5Cchi(g_4)%2C%5Cchi(g_5)%20%3D%202%2C2%2C-2%2C-w%2C-w%5E2%2C-w%5E2%2C-w.%24%24%20Also%20%24%24%5Cchi(g)%5E2%20%3D%204%2C%204%2C%200%2C%20w%5E4%2C%20w%5E2%2C%20w%5E2%2C%20w%5E4.%24%24%20With%20the%20above%20relation%20we%20have%2C%20we%20get%20%24%24%5Cchi_S(g)%20%3D%203%2C3%2C-1%2C0%2C0%2C0%2C0%24%24%20(because%20%24w%5E4%20-%20w%5E1%20%3D%200%24)%2C%20and%20%24%24%5Cchi_%7B%5CLambda%7D(g)%20%3D%201%2C1%2C1%2Cw%2Cw%5E2%2Cw%5E2%2Cw.%24%24%20We%20can%20again%20use%20the%20orthogonality%20relation%20to%20see%20they%20are%20both%20irreducible.%20So%20up%20to%20now%20we%20have%20%24%24%5Cbegin%7Barray%7D%20%7B%7Cr%7Cr%7Cr%7Cr%7Cr%7Cr%7Cr%7Cr%7C%7D%5Chline%20%20%26%20g_1%20%26%20g_2%20%26%20g_3%20%26%20g_4%20%26%20g_5%20%26%20g_6%20%26%20g_7%20%5C%5C%20%5Chline%20C_i%20%26%2024%20%26%2024%20%26%204%20%26%206%20%26%206%20%26%206%20%26%206%20%5C%5C%20%5Chline%20%5Cchi_1%20%26%201%20%26%201%20%26%201%20%26%201%20%26%201%20%26%201%20%26%201%20%5C%5C%20%5Chline%20%5Cchi%20%26%202%20%26%20-2%20%26%200%20%26%20-w%5E2%20%26%20-w%20%26%20w%20%26%20w%5E2%20%5C%5C%20%5Chline%20%5Cchi_%7B%5CLambda%7D%20%26%201%20%26%201%20%26%201%20%26%20w%20%26%20w%5E2%20%26%20w%5E2%20%26%20w%20%5C%5C%20%5Chline%20%5Cchi_S%20%26%203%20%26%203%20%26%20-1%20%26%200%20%26%200%20%26%200%20%26%200%20%5C%5C%20%5Chline%20%5Cchi_2%20%26%20%20%26%20%20%26%20%20%26%20%20%26%20%20%26%20%20%26%20%20%5C%5C%20%5Chline%20%5Cchi_3%20%26%20%20%26%20%20%26%20%20%26%20%20%26%20%20%26%20%20%26%20%20%5C%5C%20%5Chline%20%5Cchi_4%20%26%20%20%26%20%20%26%20%20%26%20%20%26%20%20%26%20%20%26%20%20%5C%5C%20%5Chline%20%5Cchi_%7Breg%7D%20%26%2024%20%26%200%20%26%200%20%26%200%20%26%200%20%26%200%20%26%200%20%5C%5C%20%5Chline%20%20%5Cend%7Barray%7D%24%24%20From%20the%20%24%5Cchi_%7Breg%7D%24%20condition%20we%20need%20%243%24%20positive%20integers%20with%20sum%20of%20squares%20%24%3D%209%24%2C%20and%20there%20is%20only%20one%20way%20to%20do%20so%2C%20namely%20%242%5E2%20%2B%202%5E2%20%2B%201%5E2%20%3D%209%24.%20So%20there%20are%20%243%24%20linear%20character%20and%20we%20can%20get%20the%20third%20one%20is%20%24%24%5Cchi_2(g)%20%3D%201%2C1%2C1%2Cw%5E2%2Cw%2Cw%2Cw%5E2%24%24%20by%20taking%20the%20'inverse'%20of%20%24%5Cchi_%7B%5CLambda%7D%24.%20Now%20we%20can%20use%20%24%24(%5Cchi_%7B%5CLambda%7D%5Ccdot%5Cchi)%20(g)%20%3D%202%2C-2%2C0%2C-1%2C-1%2C1%2C1%2C%24%24%20which%20must%20also%20be%20an%20irreducible%20character%20by%20Exercise%204.%20Similarly%20use%20%24%5Cchi_2%5Ccdot%5Cchi%24%20we%20get%20the%20last%20irreducible%20character%2C%20and%20we%20complete%20the%20table%3A%20%24%24%5Cbegin%7Barray%7D%20%7B%7Cr%7Cr%7Cr%7Cr%7Cr%7Cr%7Cr%7Cr%7C%7D%5Chline%20%20%26%20g_1%20%26%20g_2%20%26%20g_3%20%26%20g_4%20%26%20g_5%20%26%20g_6%20%26%20g_7%20%5C%5C%20%5Chline%20C_i%20%26%2024%20%26%2024%20%26%204%20%26%206%20%26%206%20%26%206%20%26%206%20%5C%5C%20%5Chline%20%5Cchi_1%20%26%201%20%26%201%20%26%201%20%26%201%20%26%201%20%26%201%20%26%201%20%5C%5C%20%5Chline%20%5Cchi%20%26%202%20%26%20-2%20%26%200%20%26%20-w%5E2%20%26%20-w%20%26%20w%20%26%20w%5E2%20%5C%5C%20%5Chline%20%5Cchi_%7B%5CLambda%7D%20%26%201%20%26%201%20%26%201%20%26%20w%20%26%20w%5E2%20%26%20w%5E2%20%26%20w%20%5C%5C%20%5Chline%20%5Cchi_S%20%26%203%20%26%203%20%26%20-1%20%26%200%20%26%200%20%26%200%20%26%200%20%5C%5C%20%5Chline%20%5Cchi_2%20%26%201%20%26%201%20%26%201%20%26%20w%5E2%20%26%20w%20%26%20w%20%26%20w%5E2%20%5C%5C%20%5Chline%20%5Cchi_3%20%26%202%20%26%20-2%20%26%200%20%26%20-1%20%26%20-1%20%26%201%20%26%201%20%5C%5C%20%5Chline%20%5Cchi_4%20%26%202%20%26%20-2%20%26%200%20%26%20-w%20%26%20-w%5E2%20%26%20w%5E2%20%26%20w%20%5C%5C%20%5Chline%20%5Cchi_%7Breg%7D%20%26%2024%20%26%200%20%26%200%20%26%200%20%26%200%20%26%200%20%26%200%20%5C%5C%20%5Chline%20%20%5Cend%7Barray%7D%24%24%0A%0A%20%20%20%20%23%23%20Exercise%208%0A%0A%20%20%20%20**Let%20%24%5Crho%24%20be%20a%20representation%20of%20%24G%24%20over%20%24%5Cmathbb%7BC%7D%24%20and%20%24%5Cdelta(g)%20%3D%20%5Cdet(%5Crho(g))%24.%20Then%20%24%5Cdelta%24%20is%20a%20linear%20character%20of%20%24G%24%20and%20%24G%2F%5Cker%7B%5Cdelta%7D%24%20is%20abelian%20(why%3F)**.%0A%0A%20%20%20%201.%20**Assume%20that%20%24%5Cdelta(g)%20%3D%20-1%24%20for%20some%20%24g%20%5Cin%20G%24.%20Show%20that%20%24G%24%20has%20a%20normal%20subgroup%20of%20index%20%242%24**%3B%0A%20%20%20%202.%20**Let%20now%20%24%5Cchi%24%20be%20a%20character%20of%20%24G%24%2C%20and%20%24h%20%5Cin%20G%24%20be%20an%20element%20of%20order%20%242%24.%20Show%20that%20either%20%24%5Cchi(h)%20%5Cequiv%20%5Cchi(1)%20%5Cpmod%7B4%7D%24%20or%20%24G%24%20has%20a%20normal%20subgroup%20of%20index%20%242%24**.%0A%0A%20%20%20%20Proof%0A%0A%20%20%20%200.%20%24%5Cdelta%24%20is%20a%20character%20because%20if%20%24hgh%5E%7B-1%7D%20%3D%20f%24%20then%20%24%5Cdelta(f)%20%3D%20%5Cdet(%5Crho(hgh%5E%7B-1%7D))%24%20%24%3D%5Cdet(%5Crho(g))%24%20%24%3D%20%5Cdelta(g)%24%2C%20so%20it%20is%20invariant%20up%20to%20conjugacy%20classes.%20It%20is%20linear%20because%20%24%5Cdelta(1)%20%3D%20%5Cdet(%5Crho(1))%20%3D%201%24.%0A%20%20%20%20%09%24G%2F%5Cker%7B%5Cdelta%7D%24%20is%20abelian%20because%20%24%5Cker%7B%5Cdelta%7D%24%20contains%20all%20the%20commutators%3A%20let%20%24x%2Cy%24%20be%20arbitrary%20element%20of%20%24G%24%2C%20then%20%24%5Cdelta(x%5E%7B-1%7Dy%5E%7B-1%7Dxy)%20%3D%20%5Cdet(%5Crho(x%5E%7B-1%7Dy%5E%7B-1%7Dxy))%24%20%24%3D%5Cdet(I)%20%3D%201%24%20because%20%24%5Crho%24%20is%20a%20homomorphism%3B%0A%20%20%20%201.%20Notice%20%24%5Cdelta%24%20is%20a%20homomorphism%20(because%20%24%5Crho%24%20is%2C%20and%20product%20of%20determinant%20is%20determinant%20of%20product).%20If%20%24%5Cdelta(g)%20%3D%20-1%24%20then%20it%20is%20an%20element%20of%20order%20%242%24%20in%20%24%5Ctext%7Bim%7D(%5Cdelta)%24%20where%20%24%5Ctext%7Bim%7D(%5Cdelta)%20%5Ccong%20G%2F%5Cker%7B%5Cdelta%7D%24%2C%20so%20%242%24%20divides%20%24%7C%20%5Ctext%7Bim%7D(%5Cdelta)%20%7C%24.%20Moreover%2C%20since%20%24%5Cdelta%24%20is%20a%20linear%20character%2C%20%24%5Ctext%7Bim%7D(%5Cdelta)%24%20is%20cyclic.%0A%20%20%20%20%09Now%20cyclic%20group%20has%20subgroup%20of%20any%20order%20that%20divides%20the%20order%20of%20the%20cyclic%20group%2C%20and%20those%20subgroups%20are%20all%20normal%2C%20in%20particular%20we%20have%20a%20normal%20subgroup%20of%20%24%5Ctext%7Bim%7D(%5Cdelta)%24%20of%20index%20%242%24%2C%20and%20by%20Correspondence%20Theorem%2C%20we%20get%20a%20normal%20subgroup%20of%20index%20%242%24%20of%20%24G%24%3B%0A%20%20%20%202.%20Suppose%20%24G%24%20does%20not%20have%20a%20subgroup%20of%20index%20%242%24%20(such%20subgroup%20is%20of%20course%20normal)%2C%20we%20want%20to%20show%20that%20we%20must%20have%20%24%5Cchi(h)%20%5Cequiv%20%5Cchi(1)%20%5Cpmod%7B4%7D%24.%0A%20%20%20%20%09By%20part%201%2C%20we%20must%20not%20have%20%24%5Cdelta(g)%20%3D%20-1%24%20for%20any%20%24g%24%2C%20but%20%241%20%3D%20%5Cdelta(h%5E2)%20%3D%20%5Cdelta(h)%5E2%24%20so%20%24%5Cdelta(h)%24%20must%20be%20%241%24.%0A%20%20%20%20%09Also%20%24%5Crho(h)%24%20either%20have%20order%20%241%24%20or%20%242%24%2C%20so%20as%20a%20matrix%20we%20must%20have%20that%20%24%5Crho(h)%24%2C%20up%20to%20conjugacy%2C%20is%20a%20diagonal%20matrix%20with%20%24%5Cpm%201%24%20on%20the%20diagonal.%20But%20we%20know%20its%20determinant%20is%20%241%24%2C%20which%20is%20the%20product%20of%20those%20%24%5Cpm%201%24's%2C%20so%20we%20must%20have%20even%20number%20%242k%24%20of%20%24-1%24's%2C%20and%20the%20rest%20%24%5Cchi(1)%20-%202k%24%20terms%20on%20the%20diagonal%20be%20%241%24%2C%20then%20a%20direct%20calculation%20gives%20us%20%24%24%5Cchi(h)%20%3D%202k(-1)%20%2B%20(%5Cchi(1)-2k)(1)%20%3D%20%5Cchi(1)%20-%204k%20%5Cequiv%20%5Cchi(1)%20%5Cpmod%7B4%7D%24%24%20as%20desired.%0A%0A%20%20%20%20%23%23%20Exercise%209%0A%0A%20%20%20%20**We%20say%20that%20%24g%20%5Cin%20G%24%20is%20real%20if%20%24%5Cchi(g)%20%5Cin%20%5Cmathbb%7BR%7D%24%20for%20all%20irreducible%20characters%20%24%5Cchi%24%20of%20%24G%24%20(or%20equivalently%2C%20%24%5Cchi(g)%20%3D%20%5Cchi(g%5E%7B-1%7D)%24).%20Prove%20that%20if%20%24%7C%20G%20%7C%24%20is%20odd%2C%20then%20no%20non-identity%20element%20of%20%24G%24%20is%20real**.%0A%0A%20%20%20%20Proof%0A%0A%20%20%20%20Suppose%20%241%20%5Cne%20g%20%5Cin%20G%24%20is%20real%2C%20then%20%24g%24%20and%20%24g%5E%7B-1%7D%24%20are%20conjugate.%20Suppose%20%24g%20%3D%20g%5E%7B-1%7D%24%2C%20then%20%24g%5E2%20%3D%201%24%3B%20since%20%24g%20%5Cne%201%24%20we%20have%20%24o(g)%20%3D%202%24%20implies%20%24%7C%20G%20%7C%24%20is%20even%2C%20so%20%24g%20%5Cne%20g%5E%7B-1%7D%24.%20Also%20we%20know%20by%20Orbit-Stabilizer%20Theorem%20that%20order%20of%20conjugacy%20class%20must%20divides%20order%20of%20%24G%24%2C%20so%20in%20our%20case%20all%20conjugacy%20classes%20have%20odd%20order.%20In%20particular%2C%20the%20conjugacy%20class%20of%20%24g%24%20cannot%20be%20just%20%24g%24%20and%20%24g%5E%7B-1%7D%24%2C%20say%20%24h%24%20is%20also%20in%20the%20conjugacy%20class.%0A%0A%20%20%20%20By%20definition%20it%20means%20there%20is%20some%20%24f%20%5Cin%20G%24%20such%20that%20%24fgf%5E%7B-1%7D%20%3D%20h%24%2C%20but%20then%20take%20inverse%20both%20sides%20we%20have%20%24fg%5E%7B-1%7Df%5E%7B-1%7D%20%3D%20h%5E%7B-1%7D%24%2C%20so%20%24h%5E%7B-1%7D%24%20is%20also%20inside%20the%20conjugacy%20class.%20%241%24%20has%20its%20own%20conjugacy%20class%2C%20so%20%24h%20%5Cne%201%24%2C%20with%20the%20same%20argument%20as%20above%20we%20see%20%24h%20%5Cne%20h%5E%7B-1%7D%24.%20This%20means%20element%20in%20conjugacy%20class%20of%20%24g%24%20always%20come%20with%20pair%2C%20so%20the%20conjugacy%20class%20of%20%24g%24%20must%20have%20even%20order%2C%20but%20as%20discussed%20above%2C%20that%20is%20impossible.%0A%20%20%20%20%22%22%22)%0A%20%20%20%20return%0A%0A%0A%40app.cell%0Adef%20_(mo)%3A%0A%20%20%20%20mo.md(r%22%22%22%0A%20%20%20%20%23%20Spring%20Semester%20Homework%203%0A%20%20%20%20%22%22%22)%0A%20%20%20%20return%0A%0A%0A%40app.cell%0Adef%20_(mo)%3A%0A%20%20%20%20mo.md(r%22%22%22%0A%20%20%20%20%23%23%20Exercise%201%20(P164.10)%0A%0A%20%20%20%20**Determine%20the%20character%20table%20of%20%24S_5%24**.%0A%0A%20%20%20%20Solution%0A%0A%20%20%20%20The%20conjugacy%20classes%20of%20%24S_n%24%20is%20determined%20by%20cycle%20types%2C%20so%20we%20have%20%24()%24%2C%20%24(12)%24%2C%20%24(123)%24%2C%20%24(1234)%24%2C%20%24(12345)%24%2C%20%24(12)(34)%24%2C%20and%20%24(123)(45)%24%2C%20in%20total%20seven%20classes%2C%20and%20it%20is%20not%20hard%20to%20calculate%20the%20size%20of%20stabilizers%3A%20%24C_i%20%3D%20120%2C%2012%2C%206%2C%204%2C%205%2C%208%2C%206%24.%20Since%20any%20permutation%20has%20the%20same%20cycle%20type%20as%20its%20inverse%2C%20we%20know%20the%20whole%20table%20is%20real.%0A%0A%20%20%20%20There%20are%20two%20representations%20%24S_n%20%5Cto%20%5Cmathbb%7BC%7D%5E%7B*%7D%24%20for%20all%20%24S_n%24%20(%24n%20%5Cge%202%24)%2C%20the%20trivial%20representation%20sending%20everything%20to%20%241%24%2C%20or%20the%20representation%20sending%20odd%20permutation%20to%20%24-1%24%20and%20even%20permutation%20to%20%241%24.%20We%20have%20%24%24%5Cbegin%7Baligned%7D%20%5Cchi_1%20%26%3D%201%2C1%2C1%2C1%2C1%2C1%2C1%3B%20%5C%5C%20%5Cchi_2%20%26%3D%201%2C-1%2C1%2C-1%2C1%2C1%2C-1.%5Cend%7Baligned%7D%24%24%20We%20also%20have%20a%20natural%20permutation%20representation%20on%20%24%5Clbrace%201%2C2%2C3%2C4%2C5%20%5Crbrace%24%2C%20gives%20a%20character%20%24%24%5Cchi_%7B%5Cpi%7D%20%3D%205%2C3%2C2%2C1%2C0%2C1%2C0.%24%24%20This%20is%20not%20an%20irreducible%20character%2C%20because%20%24%5Cfrac%7B25%7D%7B120%7D%2B%20%5Cfrac%7B9%7D%7B12%7D%20%2B%20%5Cfrac%7B4%7D%7B6%7D%20%2B%20%5Cfrac%7B1%7D%7B4%7D%20%2B%5Cfrac%7B1%7D%7B8%7D%20%3D%202%20%5Cne%201%24.%20But%20%24%5Cchi_%7B%5Cpi%7D%20-%20%5Cchi_1%24%20gives%20%244%2C2%2C1%2C0%2C-1%2C0%2C-1%24%20and%20it%20is%20an%20irreducible%20character.%20Use%20this%20times%20%24%5Cchi_2%24%20gives%20another%20irreducible%20character%2C%20because%20product%20with%20%24%5Cchi_2%24%20gives%20a%20new%20character%20with%20same%20inner%20product%20with%20itself%20as%20all%20values%20of%20%24%5Cchi_2%24%20are%20%24%5Cpm%201%24%3A%20%24%24%5Cbegin%7Baligned%7D%20%5Cchi_3%20%26%3D%204%2C2%2C1%2C0%2C-1%2C0%2C-1%3B%20%5C%5C%20%5Cchi_4%20%26%3D%204%2C-2%2C1%2C0%2C-1%2C0%2C1.%5Cend%7Baligned%7D%24%24%20Now%20we%20need%20another%20three%20characters.%20To%20match%20the%20dimension%20we%20have%20%24a%5E2%20%2B%20b%5E2%20%2B%20c%5E2%20%3D%2086%24%2C%20the%20solutions%20could%20be%20%24%5Clbrace%205%2C5%2C6%20%5Crbrace%2C%20%5Clbrace%201%2C2%2C9%20%5Crbrace%24%20or%20%24%5Clbrace%201%2C6%2C7%20%5Crbrace%24.%20Since%20the%20dimensions%20must%20divide%20%24%7C%20S%5E5%20%7C%20%3D%20120%24%2C%20the%20only%20option%20is%20to%20have%20%245%2C5%2C6%24.%20Call%20them%20%24%5Cchi_5%2C%20%5Cchi_6%2C%20%5Cchi_7%24.%0A%0A%20%20%20%20As%20mentioned%20above%2C%20the%20product%20of%20an%20irreducible%20character%20with%20%24%5Cchi_2%24%20must%20give%20another%20irreducible%20character%20(with%20the%20same%20dimension).%20Since%20there%20is%20only%20one%20character%20with%20dimension%20%246%24%2C%20%24%5Cchi_7%5Cchi_2%24%20must%20be%20%24%5Cchi_7%24%20itself%2C%20in%20particular%20%24%5Cchi_7(g)%20%3D%200%24%20if%20%24%5Cchi_2(g)%20%3D%20-1%24.%20Similarly%2C%20%24%5Cchi_5%5Cchi_2%24%20is%20either%20%24%5Cchi_5%24%20or%20%24%5Cchi_6%24%2C%20and%20%24%5Cchi_6%5Cchi_2%24%20is%20either%20%24%5Cchi_6%24%20or%20%24%5Cchi_5%24.%20However%2C%20we%20must%20have%20%24%5Cchi_5%5Cchi_2%20%3D%20%5Cchi_6%24%20and%20%24%5Cchi_6%5Cchi_2%20%3D%20%5Cchi_5%24%2C%20otherwise%20it%20forces%20%24%5Cchi_5(g)%20%3D%20%5Cchi_6(g)%20%3D%200%24%20whenever%20%24%5Cchi_2(g)%20%3D%20-1%24%2C%20but%20this%20violate%20the%20second%20orthogonality%20relation%2C%20for%20example%20consider%20%24g%20%3D%20(12)%24%2C%20then%20%24%5Csum%5Climits_t%5Cchi_t%5E2(g)%24%20%24%3D%201%2B1%2B4%2B4%24%20%24%3D%2010%20%5Cne%2012%24.%20So%20up%20to%20now%20we%20have%20%24%24%5Cbegin%7Barray%7D%20%7B%7Cr%7Cr%7Cr%7Cr%7Cr%7Cr%7Cr%7Cr%7C%7D%5Chline%20%20%26%20()%20%26%20(12)%20%26%20(123)%20%26%20(1234)%20%26%20(12345)%20%26%20(12)(34)%20%26%20(123)(45)%20%5C%5C%20%5Chline%20C_i%20%26%20120%20%26%2012%20%26%206%20%26%204%20%26%205%20%26%208%20%26%206%20%20%5C%5C%20%5Chline%20%5Cchi_1%20%26%201%20%26%201%20%26%201%20%26%201%20%26%201%20%26%201%20%26%201%20%5C%5C%20%5Chline%20%5Cchi_2%20%26%201%20%26%20-1%20%26%201%20%26%20-1%20%26%201%20%26%201%20%26%20-1%20%5C%5C%20%5Chline%20%5Cchi_3%20%26%204%20%26%202%20%26%201%20%26%200%20%26%20-1%20%26%200%20%26%20-1%20%5C%5C%20%5Chline%20%5Cchi_4%20%26%204%20%26%20-2%20%26%201%20%26%200%20%26%20-1%20%26%200%20%26%201%20%5C%5C%20%5Chline%20%5Cchi_5%20%26%205%20%26%20a%20%26%20b%20%26%20c%20%26%20d%20%26%20e%20%26%20f%20%5C%5C%20%5Chline%20%5Cchi_6%20%26%205%20%26%20-a%20%26%20b%20%26%20-c%20%26%20d%20%26%20e%20%26%20-f%20%5C%5C%20%5Chline%20%5Cchi_7%20%26%206%20%26%200%20%26%20b'%20%26%200%20%26%20d'%20%26%20e'%20%26%200%20%5C%5C%20%5Chline%20%5Cchi_%7Breg%7D%20%26%20120%20%26%200%20%26%200%20%26%200%20%26%200%20%26%200%20%26%200%20%5C%5C%20%5Chline%20%5Cend%7Barray%7D%24%24%20In%20fact%2C%20by%20above%20we%20need%20%24a%5E2%20%2B%20a%5E2%20%3D%202%24%2C%20so%20the%20only%20option%20is%20to%20have%20%24a%20%3D%20%5Cpm%201%24.%20WLOG%20say%20%24a%20%3D%201%24.%20Then%20second%20orthogonality%20relation%20on%20column%20%24(1234)%24%20with%20itself%20gives%20that%20%24c%20%3D%20%5Cpm%201%24%2C%20and%20second%20orthogonality%20relation%20on%20column%20%24(12)%24%20with%20%24(1234)%24%20gives%20that%20%24c%20%3D%20-1%24.%20Similarly%2C%20%24f%24%20must%20be%20%241%24.%20Second%20orthogonality%20relation%20of%20%24(12345)%24%20with%20itself%20shows%20that%20%24d%5E2%2Bd%5E2%2B(d')%5E2%20%3D%201%24%2C%20so%20%24d%3D%200%24%20and%20%24d'%20%3D%20%5Cpm%201%24.%20Similarly%2C%20%24b%20%3D%20%5Cpm%201%24%20and%20%24b'%20%3D%200%24%2C%20and%20that%20%24e%20%3D%20%5Cpm%201%24%20and%20%24e'%20%3D%20%5Cpm%202%24.%0A%0A%20%20%20%20(**A%20Redo**!%20The%20above%20argument%20does%20not%20work%20since%20it%20is%20not%20clear%20why%20%24d%2C%20d'%24%20must%20be%20integers.%20However%20there%20is%20a%20very%20easy%20way%20to%20do%20the%20calculation%3A%20remember%20we%20have%20a%20character%20when%20we%20introducing%20tensor%20products%20%242%5Cchi_%7B%5CLambda%7D(g)%20%3D%20%5Cchi(g)%5E2%20-%20%5Cchi(g%5E2)%24%2C%20if%20we%20use%20%24%5Cchi_3%24%20as%20our%20%24%5Cchi%24%20here%2C%20it%20won't%20be%20too%20hard%20to%20get%20%24%5Cchi_%7B%5CLambda%7D%24%20is%20%24%5Cchi_7%24%20of%20what%20we%20had.%20To%20calculate%20%24%5Cchi(g%5E2)%24%2C%20for%20example%2C%20if%20%24g%20%3D%20(123)%24%20then%20%24g%5E2%20%3D%20(123)(123)%20%3D%20(132)%24%20is%20conjugate%20to%20%24(123)%24%20so%20%24%5Cchi(g%5E2)%20%3D%201%24%2C%20etc.%20After%20this%2C%20the%20rest%20can%20be%20then%20worked%20out%20with%20the%20way%20I%20treated%20to%20%24a%2C%20c%24%20and%20%24f%24.)%0A%0A%20%20%20%20We%20have%3A%20%24%24%5Cbegin%7Barray%7D%20%7B%7Cr%7Cr%7Cr%7Cr%7Cr%7Cr%7Cr%7Cr%7C%7D%5Chline%20%20%26%20()%20%26%20(12)%20%26%20(123)%20%26%20(1234)%20%26%20(12345)%20%26%20(12)(34)%20%26%20(123)(45)%20%5C%5C%20%5Chline%20C_i%20%26%20120%20%26%2012%20%26%206%20%26%204%20%26%205%20%26%208%20%26%206%20%20%5C%5C%20%5Chline%20%5Cchi_1%20%26%201%20%26%201%20%26%201%20%26%201%20%26%201%20%26%201%20%26%201%20%5C%5C%20%5Chline%20%5Cchi_2%20%26%201%20%26%20-1%20%26%201%20%26%20-1%20%26%201%20%26%201%20%26%20-1%20%5C%5C%20%5Chline%20%5Cchi_3%20%26%204%20%26%202%20%26%201%20%26%200%20%26%20-1%20%26%200%20%26%20-1%20%5C%5C%20%5Chline%20%5Cchi_4%20%26%204%20%26%20-2%20%26%201%20%26%200%20%26%20-1%20%26%200%20%26%201%20%5C%5C%20%5Chline%20%5Cchi_5%20%26%205%20%26%201%20%26%20%5Cpm%201%20%26%20-1%20%26%200%20%26%20%5Cpm%201%20%26%201%20%5C%5C%20%5Chline%20%5Cchi_6%20%26%205%20%26%20-1%20%26%20%5Cpm%201%20%26%201%20%26%200%20%26%20%5Cpm%201%20%26%20-1%20%5C%5C%20%5Chline%20%5Cchi_7%20%26%206%20%26%200%20%26%200%20%26%200%20%26%20%5Cpm%201%20%26%20%5Cpm%202%20%26%200%20%5C%5C%20%5Chline%20%5Cchi_%7Breg%7D%20%26%20120%20%26%200%20%26%200%20%26%200%20%26%200%20%26%200%20%26%200%20%5C%5C%20%5Chline%20%20%5Cend%7Barray%7D%24%24%20Now%20the%20first%20orthogonality%20relation%20with%20%24%5Cchi_5%24%20and%20%24%5Cchi_1%24%20says%20%24%5Cfrac%7B5%7D%7B120%7D%20%2B%20%5Cfrac%7B1%7D%7B12%7D%20%2B%20%5Cfrac%7B%5Cpm%201%7D%7B6%7D%20-%20%5Cfrac%7B1%7D%7B4%7D%20%2B%20%5Cfrac%7B%5Cpm%201%7D%7B8%7D%20%2B%20%5Cfrac%7B1%7D%7B6%7D%20%3D%200%24%2C%20so%20we%20must%20have%20%24-%20%5Cfrac%7B1%7D%7B6%7D%24%20and%20%24%2B%5Cfrac%7B1%7D%7B8%7D%24.%20Similarly%20the%20first%20orthogonality%20relation%20with%20%24%5Cchi_7%24%20and%20%24%5Cchi_1%24%20gives%20the%20last%20information%20we%20need%2C%20and%20we%20have%3A%20%24%24%5Cbegin%7Barray%7D%20%7B%7Cr%7Cr%7Cr%7Cr%7Cr%7Cr%7Cr%7Cr%7Cr%7C%7D%5Chline%20%20%26%20()%20%26%20(12)%20%26%20(123)%20%26%20(1234)%20%26%20(12345)%20%26%20(12)(34)%20%26%20(123)(45)%20%5C%5C%20%5Chline%20C_i%20%26%20120%20%26%2012%20%26%206%20%26%204%20%26%205%20%26%208%20%26%206%20%20%5C%5C%20%5Chline%20%5Cchi_1%20%26%201%20%26%201%20%26%201%20%26%201%20%26%201%20%26%201%20%26%201%20%5C%5C%20%5Chline%20%5Cchi_2%20%26%201%20%26%20-1%20%26%201%20%26%20-1%20%26%201%20%26%201%20%26%20-1%20%5C%5C%20%5Chline%20%5Cchi_3%20%26%204%20%26%202%20%26%201%20%26%200%20%26%20-1%20%26%200%20%26%20-1%20%5C%5C%20%5Chline%20%5Cchi_4%20%26%204%20%26%20-2%20%26%201%20%26%200%20%26%20-1%20%26%200%20%26%201%20%5C%5C%20%5Chline%20%5Cchi_5%20%26%205%20%26%201%20%26%20-1%20%26%20-1%20%26%200%20%26%201%20%26%201%20%5C%5C%20%5Chline%20%5Cchi_6%20%26%205%20%26%20-1%20%26%20-1%20%26%201%20%26%200%20%26%201%20%26%20-1%20%5C%5C%20%5Chline%20%5Cchi_7%20%26%206%20%26%200%20%26%200%20%26%200%20%26%201%20%26%20-%202%20%26%200%20%5C%5C%20%5Chline%20%5Cchi_%7Breg%7D%20%26%20120%20%26%200%20%26%200%20%26%200%20%26%200%20%26%200%20%26%200%20%5C%5C%20%5Chline%20%20%5Cend%7Barray%7D%24%24%0A%0A%20%20%20%20%23%23%20Exercise%202%0A%0A%20%20%20%20**Look%20up%20the%20character%20table%20of%20%24S_6%24%2C%20now%20recover%20the%20character%20table%20of%20%24A_6%24%20from%20it**.%0A%0A%20%20%20%20Solution%0A%0A%20%20%20%20Here%20is%20the%20character%20table%20for%20%24S_6%24%3A%20%24%24%5Cbegin%7Barray%7D%20%7B%7Cr%7Cr%7Cr%7Cr%7Cr%7Cr%7Cr%7Cr%7Cr%7Cr%7Cr%7Cr%7C%7D%5Chline%20%5Ctext%7Bcycle%20type%7D%20%26%200%20%26%202%20%26%20222%20%26%203%20%26%2033%20%26%2022%20%26%204%20%26%2024%20%26%2023%20%26%206%20%26%205%20%5C%5C%20%5Chline%20C_i%20%26%20720%20%26%2048%20%26%2048%20%26%2018%20%26%2018%20%26%2016%20%26%208%20%26%208%20%26%206%20%26%206%20%26%205%20%5C%5C%20%5Chline%20%5Cchi_1%20%26%201%20%26%201%20%26%201%20%26%201%20%26%201%20%26%201%20%26%201%20%26%201%20%26%201%20%26%201%20%26%201%20%5C%5C%20%5Chline%20%5Cchi_2%20%26%201%20%26%20-1%20%26%20-1%20%26%201%20%26%201%20%26%201%20%26%20-1%20%26%201%20%26%20-1%20%26%20-1%20%26%201%20%5C%5C%20%5Chline%20%5Cchi_3%20%26%205%20%26%203%20%26%20-1%20%26%202%20%26%20-1%20%26%201%20%26%201%20%26%20-1%20%26%200%20%26%20-1%20%26%200%20%5C%5C%20%5Chline%20%5Cchi_4%20%26%205%20%26%20-3%20%26%201%20%26%202%20%26%20-1%20%26%201%20%26%20-1%20%26%20-1%20%26%200%20%26%201%20%26%200%20%5C%5C%20%5Chline%20%5Cchi_5%20%26%205%20%26%20-1%20%26%203%20%26%20-1%20%26%202%20%26%201%20%26%201%20%26%20-1%20%26%20-1%20%26%200%20%26%200%20%5C%5C%20%5Chline%20%5Cchi_6%20%26%205%20%26%201%20%26%20-3%20%26%20-1%20%26%202%20%26%201%20%26%20-1%20%26%20-1%20%26%201%20%26%200%20%26%200%20%5C%5C%20%5Chline%20%5Cchi_7%20%26%209%20%26%203%20%26%203%20%26%200%20%26%200%20%26%201%20%26%20-1%20%26%201%20%26%200%20%26%200%20%26%20-1%20%5C%5C%20%5Chline%20%5Cchi_8%20%26%209%20%26%20-3%20%26%20-3%20%26%200%20%26%200%20%26%201%20%26%201%20%26%201%20%26%200%20%26%200%20%26%20-1%20%5C%5C%20%5Chline%20%5Cchi_9%20%26%2010%20%26%202%20%26%20-2%20%26%201%20%26%201%20%26%20-2%20%26%200%20%26%200%20%26%20-1%20%26%201%20%26%200%20%5C%5C%20%5Chline%20%5Cchi_%7B10%7D%20%26%2010%20%26%20-2%20%26%202%20%26%201%20%26%201%20%26%20-2%20%26%200%20%26%200%20%26%201%20%26%20-1%20%26%200%20%5C%5C%20%5Chline%20%5Cchi_%7B11%7D%20%26%2016%20%26%200%20%26%200%20%26%20-2%20%26%20-2%20%26%200%20%26%200%20%26%200%20%26%200%20%26%200%20%26%201%20%5C%5C%20%5Chline%20%20%5Cend%7Barray%7D%24%24%20The%20alternating%20subgroup%20has%20%247%24%20conjugacy%20classes%2C%20which%20are%20of%20cycle%20type%20%240%24%2C%20%2422%24%2C%20%243%24%2C%20%2433%24%2C%20%2424%24%2C%20and%20two%20half%20of%20%245%24%20(for%20example%2C%20%24(12345)%24%20and%20%24(13524)%24%20are%20not%20conjugate%20inside%20%24A_6%24).%20Much%20of%20the%20information%20we%20need%20are%20just%20inherited%20from%20%24S_6%24%3A%20%24%24%5Cbegin%7Barray%7D%20%7B%7Cr%7Cr%7Cr%7Cr%7Cr%7Cr%7Cr%7Cr%7C%7D%5Chline%20%5Ctext%7Bcycle%20type%7D%20%26%200%20%26%203%20%26%2033%20%26%2022%20%26%2024%20%26%205a%20%26%205b%20%5C%5C%5Chline%20C_i%20%26%20360%20%26%209%20%26%209%20%26%208%20%26%204%20%26%205%20%26%205%20%20%5C%5C%20%5Chline%20%5Cchi_1%20%26%201%20%26%201%20%26%201%20%26%201%20%26%201%20%26%201%20%26%201%5C%5C%20%5Chline%20%5Cchi_3%20%26%205%20%26%202%20%26%20-1%20%26%201%20%26%20-1%20%26%200%20%26%200%5C%5C%20%5Chline%20%5Cchi_5%20%26%205%20%26%20-1%20%26%202%20%26%201%20%26%20-1%20%26%200%20%26%200%5C%5C%20%5Chline%20%5Cchi_7%20%26%209%20%26%200%20%26%200%20%26%201%20%26%201%20%26%20-1%20%26%20-1%5C%5C%20%5Chline%20%5Cchi_%7B9%7D%20%26%2010%20%26%201%20%26%201%20%26%20-2%20%26%200%20%26%200%20%26%200%5C%5C%20%5Chline%20%5Cchi_%7B11a%7D%20%26%208%20%26%20a%20%26%20b%20%26%20c%20%26%20d%20%26%20e%20%26%20f%20%5C%5C%20%5Chline%20%5Cchi_%7B11b%7D%20%26%208%20%26%20a'%20%26%20b'%20%26%20c'%20%26%20d'%20%26%20e'%20%26%20f'%20%5C%5C%20%5Chline%20%20%5Cend%7Barray%7D%24%24%20We%20get%20two%20%248%24-dimension%20modules%20because%20that%20is%20the%20only%20integers%20matches%20%24a%5E2%20%2B%20b%5E2%20%3D%20128%24.%20Also%20remember%20this%20table%20should%20be%20all%20real.%20We%20can%20get%20%24c%2C%20c'%2C%20d%2C%20d'%20%3D%200%24%20because%20the%20second%20orthogonality%20relation%20(%241%2B1%2B1%2B1%2B4%20%3D%208%24%20and%20%241%2B1%2B1%2B1%20%3D%204%24%20already).%20We%20use%20the%20orthogonal%20relations%20to%20solve%20the%20rest.%0A%0A%20%20%20%201.%20For%20%24a%24%20and%20%24a'%24%2C%20we%20have%20%24a%5E2%20%2B%20%7Ba'%7D%5E2%20%2B%201%20%2B%201%20%2B%204%20%2B%201%20%3D%209%24%2C%20so%20%24a%5E2%20%2B%20%7Ba'%7D%5E2%20%3D%202%24%2C%20and%20that%20%241%2B10-5%2B10%2B8a%2B8a'%20%3D%200%24%20so%20%24a%2Ba'%20%3D%20-2%24%2C%20so%20there%20is%20only%20one%20solution%20that%20%24a%20%3D%20a'%20%3D%20-1%24%3B%0A%20%20%20%202.%20For%20%24b%24%20and%20%24b'%24%2C%20we%20get%20%24b%5E2%20%2B%20%7Bb'%7D%5E2%20%3D%202%24%20and%20%24b%20%2B%20b'%20%3D%20-2%24%20as%20well%2C%20so%20%24b%20%3D%20b'%20%3D%20-1%24%3B%0A%20%20%20%203.%20For%20%24e%24%20and%20%24e'%24%2C%20we%20get%20%24e%5E2%20%2B%20%7Be'%7D%5E2%20%3D%203%24%20and%20%24e%20%2B%20e'%20%3D%201%24%2C%20and%20we%20get%20the%20same%20equation%20for%20%24f%24%20and%20%24f'%24%2C%20since%20no%20rows%20are%20the%20same%2C%20we%20get%20%24e%2C%20e'%20%3D%20%5Cfrac%7B1%20%5Cpm%20%5Csqrt%7B5%7D%7D%7B2%7D%24%20and%20%24f%2C%20f'%20%3D%20%5Cfrac%7B1%20%5Cmp%20%5Csqrt%7B5%7D%7D%7B2%7D%24%20and%20we%20have%20complete%20the%20table%3A%20%24%24%5Cbegin%7Barray%7D%20%7B%7Cr%7Cr%7Cr%7Cr%7Cr%7Cr%7Cr%7Cr%7C%7D%5Chline%20%5Ctext%7Bcycle%20type%7D%20%26%200%20%26%203%20%26%2033%20%26%2022%20%26%2024%20%26%205a%20%26%205b%20%5C%5C%5Chline%20C_i%20%26%20360%20%26%209%20%26%209%20%26%208%20%26%204%20%26%205%20%26%205%20%20%5C%5C%20%5Chline%20%5Cchi_1%20%26%201%20%26%201%20%26%201%20%26%201%20%26%201%20%26%201%20%26%201%5C%5C%20%5Chline%20%5Cchi_3%20%26%205%20%26%202%20%26%20-1%20%26%201%20%26%20-1%20%26%200%20%26%200%5C%5C%20%5Chline%20%5Cchi_5%20%26%205%20%26%20-1%20%26%202%20%26%201%20%26%20-1%20%26%200%20%26%200%5C%5C%20%5Chline%20%5Cchi_7%20%26%209%20%26%200%20%26%200%20%26%201%20%26%201%20%26%20-1%20%26%20-1%5C%5C%20%5Chline%20%5Cchi_%7B9%7D%20%26%2010%20%26%201%20%26%201%20%26%20-2%20%26%200%20%26%200%20%26%200%5C%5C%20%5Chline%20%5Cchi_%7B11a%7D%20%26%208%20%26%20-1%20%26%20-1%20%26%200%20%26%200%20%26%20%5Cfrac%7B1%2B%5Csqrt%7B5%7D%7D%7B2%7D%20%26%20%5Cfrac%7B1-%5Csqrt%7B5%7D%7D%7B2%7D%20%5C%5C%20%5Chline%20%5Cchi_%7B11b%7D%20%26%208%20%26%20-1%20%26%20-1%20%26%200%20%26%200%20%26%20%5Cfrac%7B1-%5Csqrt%7B5%7D%7D%7B2%7D%20%26%20%5Cfrac%7B1%2B%5Csqrt%7B5%7D%7D%7B2%7D%20%5C%5C%20%5Chline%20%5Cend%7Barray%7D%24%24%0A%0A%20%20%20%20%23%23%20Exercise%203%20(P164.8)%0A%0A%20%20%20%20**Show%20that%20if%20%24G%24%20acts%20doubly%20transitively%20on%20%24X%24%2C%20then%20the%20character%20of%20%24%5Cmathbb%7BC%7DX%24%20is%20the%20sum%20of%20the%20principal%20character%20and%20exactly%20one%20other%20irreducible%20character**.%0A%0A%20%20%20%20Proof%0A%0A%20%20%20%20Let's%20restate%20things%20to%20get%20a%20bit%20clearer%3A%20suppose%20%24G%24%20acts%20doubly%20transitively%20on%20%24X%24%20(corresponds%20to%20a%20permutation%20module)%2C%20let%20%24%5Cchi%24%20be%20the%20corresponding%20character.%20The%20action%20of%20%24G%24%20on%20%24X%24%20induces%20the%20action%20%24G%24%20on%20%24X%20%5Ctimes%20X%24%20by%20%24g(x%2Cy)%20%3D%20(gx%2C%20gy)%24.%20Let%20us%20denote%20the%20character%20corresponds%20to%20this%20action%20by%20%24%5Cchi_d%24.%0A%0A%20%20%20%20First%20notice%20that%20%24%5Cchi_d%20%3D%20%5Cchi%5E2%24%2C%20this%20can%20be%20seen%20by%20counting%20the%20fixed%20point%20of%20the%20action%20because%20we%20have%20permutation%20modules%3A%20if%20%24g%24%20acts%20on%20%24X%24%20with%20%24n%24%20fixed%20points%20%24x_1%2C%5Cdots%2Cx_n%24%2C%20then%20%24g%24%20acts%20on%20%24X%20%5Ctimes%20X%24%20has%20exactly%20%24n%20%5Ctimes%20n%24%20fixed%20points%20%24(x_1%2C%20x_1)%2C(x_1%2Cx_2)%2C%5Cdots%2C(x_1%2Cx_n)%2C%5Cdots%2C(x_n%2Cx_n)%24.%0A%0A%20%20%20%20By%20the%20discussion%20in%20the%20book%2C%20we%20know%20that%20%24%5Clangle%20%5Cchi_d%2C%20%5Cchi_1%20%5Crangle%24%20equals%20the%20number%20of%20orbits%20of%20%24X%20%5Ctimes%20X%24%20under%20the%20action%20of%20%24G%24%2C%20and%20by%20definition%20of%20double-transitivity%2C%20there%20are%20exactly%20%242%24%20orbits%2C%20namely%20the%20diagonal%20and%20everything%20else%2C%20so%20%24%5Clangle%20%5Cchi_d%2C%20%5Cchi_1%20%5Crangle%20%3D%202%24.%20Since%20%24%5Cchi_d%20%3D%20%5Cchi%5E2%24%2C%20we%20get%20%24%5Clangle%20%5Cchi_d%2C%20%5Cchi_1%20%5Crangle%20%3D%20%5Clangle%20%5Cchi%2C%5Cchi%20%5Crangle%24%20(because%20those%20are%20all%20integers)%20%24%3D%202%24.%20In%20particular%20we%20must%20have%20%24%5Clangle%20%5Cchi%2C%20%5Cchi_1%20%5Crangle%20%3D%201%24%2C%20thus%20we%20must%20have%20%24%5Clangle%20%5Cchi%2C%5Cchi'%20%5Crangle%20%3D%201%24%20for%20another%20irreducible%20character%20%24%5Cchi'%24%2C%20and%20%24%5Clangle%20%5Cchi%2C%5Cchi''%20%5Crangle%20%3D%200%24%20for%20any%20other%20irreducible%20character.%0A%0A%20%20%20%20It%20follows%20that%20the%20module%20corresponds%20to%20%24%5Cchi%24%20is%20a%20direct%20sum%20of%20trivial%20module%20and%20a%20simple%20module%2C%20and%20%24%5Cchi%20%3D%20%5Cchi_1%20%2B%20%5Cchi'%24%2C%20a%20sum%20of%20principal%20character%20and%20another%20irreducible%20character.%0A%0A%20%20%20%20%23%23%20Exercise%204%0A%0A%20%20%20%20**Let%20%24G%24%20be%20a%20group%20of%20odd%20order%20%24n%24%2C%20%24%5Cchi_1%24%20be%20the%20trivial%20character%20of%20%24G%24%2C%20and%20%24%5Ceta%24%20be%20an%20irreducible%20real%20character%20of%20%24G%24%2C%20i.e.%20%24%5Ceta%20%3D%20%5Coverline%7B%5Ceta%7D%24**.%0A%0A%20%20%20%201.%20**Check%20that**%20%24%24%5Clangle%20%5Ceta%2C%5Cchi_1%20%5Crangle%20%3D%20%5Cfrac%7B1%7D%7Bn%7D(%5Ceta(1)%20%2B%202%5Calpha)%24%24%20**where%20%24%5Calpha%24%20is%20an%20algebraic%20integer**%3B%0A%20%20%20%202.%20**Deduce%20that%20%24%5Ceta%20%3D%20%5Cchi_1%24**.%0A%0A%20%20%20%20Solution%0A%0A%20%20%20%201.%20If%20%24%5Ceta%20%3D%20%5Cchi_1%24%20then%20the%20statement%20is%20obvious%3A%20%24%5Ceta(1)%20%3D%201%24%2C%20%24n%24%20is%20odd%2C%20thus%20if%20we%20take%20%24%5Calpha%20%3D%20(n-1)%2F2%24%20then%20it%20is%20an%20integer%20thus%20an%20algebraic%20integer%2C%20and%20we%20have%20%24%24%5Clangle%20%5Ceta%2C%5Cchi_1%20%5Crangle%20%3D%201%20%3D%20%5Cfrac%7B1%7D%7Bn%7D(1%20%2B%202%5Cfrac%7Bn-1%7D%7B2%7D).%24%24%20Suppose%20now%20%24%5Ceta%20%5Cne%20%5Cchi_1%24.%20From%20orthogonality%20relation%20we%20know%20that%20%24%5Clangle%20%5Ceta%2C%5Cchi_1%20%5Crangle%20%3D%200%24.%20Expand%20LHS%20we%20have%20that%20%24%240%20%3D%20%5Cfrac%7B1%7D%7Bn%7D%5Csum_%7Bg%5Cin%20G%7D%5Ceta(g)%5Coverline%7B%5Cchi_1(g)%7D%20%3D%20%5Cfrac%7B1%7D%7Bn%7D%5Csum_%7Bg%5Cin%20G%7D%5Ceta(g).%24%24%20In%20particular%20%24%24%5Csum_%7Bg%20%5Cin%20G%7D%5Ceta(g)%20%3D%20%5Ceta(1)%20%2B%20%5Csum_%7Bg%20%5Cne%201%7D%5Ceta(g)%20%3D%200.%24%24%20Now%20from%20before%20we%20know%20if%20%24%7C%20G%20%7C%24%20is%20odd%20then%20%24g%20%5Cne%20g%5E%7B-1%7D%24%20for%20all%20%24g%20%5Cne%201%24%20(otherwise%20this%20element%20has%20order%20%242%24%2C%20contradict%20that%20%24%7C%20G%20%7C%24%20is%20odd)%2C%20also%20we%20know%20that%20%24%5Ceta(g%5E%7B-1%7D)%24%20%24%3D%20%5Coverline%7B%5Ceta(g)%7D%24%20%24%3D%20%5Ceta(g)%24%20by%20assumption%2C%20so%20we%20have%20%24%24%5Cbegin%7Balign*%7D0%20%26%3D%20%5Ceta(1)%20%2B%20%5Csum_%7B%5Clbrace%20g%2C%20g%5E%7B-1%7D%20%5Crbrace%7D%5Ceta(g)%20%2B%20%5Ceta(g%5E%7B-1%7D)%20%5C%5C%200%26%3D%5Ceta(1)%20%2B%202%20%5Csum_%7B%5Clbrace%20g%2C%20g%5E%7B-1%7D%20%5Crbrace%7D%5Ceta(g)%20%5Ctag%7B1%7D%20%5C%5C%20-%5Cfrac%7B%5Ceta(1)%7D%7B2%7D%20%26%3D%20%5Csum_%7B%5Clbrace%20g%2C%20g%5E%7B-1%7D%20%5Crbrace%7D%5Ceta(g).%20%5Cend%7Balign*%7D%24%24%20The%20LHS%20is%20a%20fraction%2C%20because%20we%20know%20dimension%20of%20module%20must%20divide%20order%20of%20%24G%24%20so%20%24%5Ceta(1)%24%20is%20odd.%20But%20a%20fraction%20is%20not%20an%20algebraic%20integer%2C%20yet%20the%20RHS%20is%20a%20sum%20of%20algebraic%20integers%20thus%20an%20algebraic%20integer%2C%20so%20we%20have%20a%20contradiction.%20So%20%24%5Ceta%24%20must%20be%20%24%5Cchi_1%24%20and%20the%20statement%20is%20true%20as%20discussed.%0A%20%20%20%20%20%20%20%20If%20we%20just%20want%20to%20verify%20the%20statement%2C%20then%20%24(1)%24%20gives%20the%20equation%3B%0A%20%20%20%202.%20Done%20above.%0A%0A%20%20%20%20%23%23%20Exercise%205%0A%0A%20%20%20%20**Let%20%24R%20%5Csubseteq%20S%20%5Csubseteq%20T%24%20be%20domains%2C%20suppose%20all%20elements%20of%20%24S%24%20are%20%24R%24-integral%20and%20%24t%20%5Cin%20T%24%20is%20%24S%24-integral.%20Show%20that%20%24t%24%20is%20%24R%24-integral**.%0A%0A%20%20%20%20Proof%0A%0A%20%20%20%20Suppose%20%24t%20%5Cin%20T%24%20is%20an%20%24S%24-integer.%20By%20definition%2C%20%24t%5En%20%2B%20s_1%20t%5E%7Bn-1%7D%20%2B%20%5Cdots%20%2B%20s_n%20%3D%200%24%20where%20%24s_i%20%5Cin%20S%24.%20In%20particular%2C%20%24t%24%20is%20integer%20over%20%24R%5Bs_1%2C%20%5Cdots%2C%20s_n%5D%24%2C%20which%20means%20%24R%5Bs_1%2C%5Cdots%2C%20s_n%2C%20t%5D%24%20is%20a%20finitely%20generated%20%24R%5Bs_1%2C%5Cdots%2Cs_n%5D%24-module.%20On%20the%20other%20hand%2C%20%24R%5Bs_1%2C%20%5Cdots%2C%20s_n%5D%24%20is%20a%20finitely%20generated%20%24R%24-module%20because%20%24s_1%2C%5Cdots%2Cs_n%24%20are%20by%20assumption%20%24R%24-integers.%20Thus%20%24R%5Bs_1%2C%5Cdots%2C%20s_n%2C%20t%5D%24%20is%20a%20finitely%20generated%20%24R%24-module%2C%20and%20thus%20%24t%24%20is%20an%20%24R%24-integer.%0A%0A%20%20%20%20%23%23%20Exercise%206%0A%0A%20%20%20%20**Let%20%24a%20%5Cin%20%5Cmathbb%7BR%7D%24%20be%20a%20root%20of%20%24x%5E3%20%2B%206x%20%2B%2034%20%5Cin%20%5Cmathbb%7BZ%7D%5Bx%5D%24%20and%20%24R%20%3D%20%5Cmathbb%7BZ%7D%5Ba%5D%24.%20Show%20that%20%24R%24%20is%20not%20integrally%20closed%20(look%20at%20%24%5Cfrac%7B1%2Ba%7D%7B3%7D%24)**.%0A%0A%20%20%20%20Solution%0A%0A%20%20%20%20Notice%20if%20%24a%5E3%20%2B%206a%20%2B%2034%20%3D%200%24%2C%20then%20%24%2427(%5Cfrac%7Ba%2B1%7D%7B3%7D)%5E3%20-%2027%20(%5Cfrac%7Ba%2B1%7D%7B3%7D)%5E2%20%2B%2027(%5Cfrac%7Ba%2B1%7D%7B3%7D)%20%2B%2027%20%3D%20a%5E3%2B6a%2B34%20%3D%200%2C%24%24%20so%20%24%5Cfrac%7B1%2Ba%7D%7B3%7D%24%20is%20a%20%24%5Cmathbb%7BZ%7D%5Ba%5D%24-integer.%20But%20it%20is%20not%20contained%20in%20%24%5Cmathbb%7BZ%7D%5Ba%5D%24%3A%0A%0A%20%20%20%20Suppose%20it%20is%2C%20then%20%24%5Cfrac%7B1%2Ba%7D%7B3%7D%20%3D%20b_0%20%2B%20b_1%20a%20%2B%20%5Cdots%20%2B%20b_na%5En%24%20for%20some%20%24n%24%20and%20%24b_i%20%5Cin%20%5Cmathbb%7BZ%7D%24%2C%20then%20%24(3b_0%20-%201)%20%2B%20(3b_1%20-%201)a%20%2B%20%5Cdots%20%2B%203b_na%5En%20%3D%200%24.%20Since%20%24a%5E3%2B6a%2B34%24%20is%20irreducible%2C%20it%20must%20be%20a%20factor%20of%20the%20above%20polynomial%2C%20but%20the%20coefficients%20do%20not%20match%20modulo%20%243%24%3A%20say%20%24%24(a%5E3%2B6a%2B34)(c_ma%5Em%20%2B%20%5Cdots%20%2Bc_1a%5E1%20%2B%20c_0)%20%3D%20(3b_0%20-%201)%20%2B%20(3b_1%20-%201)a%20%2B%20%5Cdots%20%2B%203b_na%5En%2C%24%24%20then%20%24c_m%20%5Cequiv%200%20%5Cpmod%7B3%7D%24%20as%20the%20leading%20coefficient%20RHS%20is%20%243b_n%24%2C%20and%20similar%20for%20all%20%24c_i%24's%2C%20in%20particular%20%24c_0%20%5Cequiv%200%20%5Cpmod%7B3%7D%24%2C%20but%20then%20the%20constant%20term%20on%20LHS%20%24%5Cequiv%200%20%5Cpmod%7B3%7D%24%2C%20but%20%24%5Cequiv%202%20%5Cpmod%7B3%7D%24%20on%20RHS.%0A%0A%20%20%20%20%23%23%20Exercise%207%0A%0A%20%20%20%20**Let%20%24I%20%3D%20(y%5E2%20-%20x%5E3)%24%20be%20an%20ideal%20in%20the%20polynomial%20ring%20%24R%20%3D%20%5Cmathbb%7BC%7D%5Bx%2C%20y%5D%24%20and%20%24S%20%3D%20R%2FI%24**.%0A%0A%20%20%20%201.%20**Is%20%24S%24%20a%20domain%3F%20What%20does%20the%20answer%20tell%20us%20about%20the%20ideal%20%24I%24**%3F%20*Hint%3A%20Consider%20the%20map%20%24f(x%2Cy)%20%5Cto%20f(t%5E2%2C%20t%5E3)%20%5Cin%20%5Cmathbb%7BC%7D%5Bt%5D%24%20and%20determine%20its%20kernel*.%0A%20%20%20%202.%20**Show%20that%20%24S%24%20is%20not%20integrally%20closed%20(and%20thus%20not%20a%20UFD)**.%0A%0A%20%20%20%20Solution%0A%0A%20%20%20%201.%20Consider%20the%20map%20%24%5Cvarphi%3A%20%5Cmathbb%7BC%7D%5Bx%2Cy%5D%20%5Cto%20%5Cmathbb%7BC%7D%5Bt%5D%24%20by%20%24f(x%2Cy)%20%5Cmapsto%20f(t%5E2%2C%20t%5E3)%24.%20Any%20%24f(x%2Cy)%24%20can%20be%20viewed%20as%20a%20polynomial%20in%20%24y%24%2C%20also%20%24y%5E2%20-%20x%5E3%24%20can%20be%20viewed%20as%20a%20polynomial%20in%20%24y%24%2C%20so%20we%20can%20perform%20an%20Euclidean%20division%2C%20say%20%24%24f(x%2Cy)%20%3D%20(y%5E2%20-%20x%5E3)g(x%2Cy)%20%2B%20r(x%2Cy)%2C%24%24%20where%20the%20degree%20of%20%24y%24%20in%20%24r(x%2Cy)%24%20is%20at%20most%20%241%24%2C%20so%20we%20can%20write%20%24%24f(x%2Cy)%20%3D%20(y%5E2-x%5E3)g(x%2Cy)%20%2B%20r_1(x)%20%2B%20r_2(x)y.%24%24%20The%20image%20of%20this%20polynomial%20under%20%24%5Cvarphi%24%20is%20%24%24f(t%5E2%2Ct%5E3)%20%3D%20(t%5E6%20-%20t%5E6)g(t%5E2%2C%20t%5E3)%20%2B%20r_1(t%5E2)%20%2B%20r_2(t%5E2)t%5E3%20%3D%20r_1(t%5E2)%20%2B%20r_2(t%5E2)t%5E3.%24%24%20Notice%20now%20%24r_1(t%5E2)%24%20only%20contains%20even%20degree%20monomials%2C%20yet%20%24r_2(t%5E2)t%5E3%24%20only%20contains%20odd%20degree%20monomials%2C%20so%20they%20don't%20'interact'%20(in%20particular%2C%20cancel%20terms%20when%20being%20added%2C%20this%20argument%20works%20as%20long%20as%20the%20powers%20of%20%24x%2C%20y%24%20are%20coprime)%20with%20each%20other%2C%20so%20their%20summation%20is%20%240%24%20if%20and%20only%20if%20all%20coefficients%20of%20%24r_1%24%20and%20%24r_2%24%20are%20%240%24.%20This%20means%20the%20kernel%20of%20%24%5Cvarphi%24%20consists%20exactly%20of%20those%20%24f%24%20such%20that%20%24f(x%2Cy)%20%3D%20(y%5E2-x%5E3)g(x%2Cy)%24%2C%20in%20other%20words%2C%20%24%24%5Cker%7B%5Cvarphi%7D%20%3D%20(y%5E2-x%5E3)%2C%24%24%20and%20isomorphism%20theorem%20tells%20us%20%24S%20%3D%20%5Cmathbb%7BC%7D%5Bx%2Cy%5D%2F%5Cker%7B%5Cvarphi%7D%20%5Ccong%20im%7B%5Cvarphi%7D%20%5Csubset%20%5Cmathbb%7BC%7D%5Bt%5D%24.%20Since%20%24%5Cmathbb%7BC%7D%24%20is%20a%20field%2C%20it%20is%20a%20domain%2C%20thus%20%24%5Cmathbb%7BC%7D%5Bt%5D%24%20is%20a%20domain%20(Gauss)%2C%20thus%20%24im%7B%5Cvarphi%7D%24%20is%20an%20ideal%20of%20%24%5Cmathbb%7BC%7D%5Bt%5D%24%20is%20a%20domain%2C%20thus%20%24S%24%20is%20a%20domain.%20Also%20we%20know%20that%20%24R%2FI%24%20is%20a%20domain%20if%20and%20only%20if%20%24I%24%20is%20prime%2C%20so%20%24I%24%20is%20a%20prime%20ideal.%0A%20%20%20%202.%20Consider%20%24s%20%3D%20%5Cfrac%7By%7D%7Bx%7D%24%2C%20clearly%20it%20is%20not%20an%20element%20of%20%24S%24.%20But%20%24s%5E2%20%3D%20%5Cfrac%7By%5E2%7D%7Bx%5E2%7D%20%3D%20%5Cfrac%7Bx%5E3%7D%7Bx%5E2%7D%20%3D%20x%24%20so%20%24s%5E2%20-%20x%20%3D%200%24%20in%20%24S%24%2C%20which%20means%20%24s%24%20is%20an%20%24S%24-integer.%0A%0A%20%20%20%20%23%23%20Exercise%208%0A%0A%20%20%20%20**Let**%20%24%24%5Cchi_n%20%3D%20%5Cfrac%7B%5Csin(n%2B1)%5Ctheta%7D%7B%5Csin(%5Ctheta)%7D%2C%20%5Cchi_m%20%3D%20%5Cfrac%7B%5Csin(m%2B1)%5Ctheta%7D%7B%5Csin(%5Ctheta)%7D%24%24%20**be%20the%20characters%20of%20the%20irreducible%20representations%20%24%5Cvarphi_n%24%20and%20%24%5Cvarphi_m%24%20of%20%24SU(2)%24%2C%20%24m%20%5Cge%20n%20%5Cge%201%24.%20Decompose%20the%20product%20%24%5Cchi_n%5Cchi_m%24%20into%20sum%20of%20characters%20of%20%24SU(2)%24.%20What%20kind%20of%20information%20is%20encoded%20in%20this%20decomposition**%3F%0A%0A%20%20%20%20Solution%0A%0A%20%20%20%20We%20may%20write%20%24%5Cchi_n%24%20and%20%24%5Cchi_m%24%20as%20%24%24%5Cchi_n(z_%7B%5Ctheta%7D)%20%3D%20e%5E%7Bin%5Ctheta%7D%20%2B%20e%5E%7Bi(n-2)%5Ctheta%7D%20%2B%20%5Cdots%20%2B%20e%5E%7B-in%5Ctheta%7D%20%3D%20%5Csum_%7Bk%3D0%7D%5Ene%5E%7Bi(n-2k)%5Ctheta%7D%24%24%20and%20similar%20for%20%24%5Cchi_m(z_%7B%5Ctheta%7D)%24.%20Multiply%20them%20together%2C%20we%20get%20%24%24%5Cbegin%7Baligned%7D(%5Csum_%7Bk%3D0%7D%5Ene%5E%7Bi(n-2k)%5Ctheta%7D)(%5Csum_%7Bj%3D0%7D%5Eme%5E%7Bi(m-2j)%5Ctheta%7D)%20%26%3D%20e%5E%7Bin%5Ctheta%7D(e%5E%7Bim%5Ctheta%7D%20%2B%20e%5E%7Bi(m-2)%5Ctheta%7D%20%2B%20%5Cdots%20%2B%20e%5E%7B-im%5Ctheta%7D)%5C%5C%20%26%2B%20e%5E%7Bi(n-2)%5Ctheta%7D(e%5E%7Bim%5Ctheta%7D%20%2B%20e%5E%7Bi(m-2)%5Ctheta%7D%20%2B%20%5Cdots%20%2B%20e%5E%7B-im%5Ctheta%7D)%5C%5C%20%26%2B%20%5Cdots%20%5C%5C%20%26%2Be%5E%7B-in%5Ctheta%7D(e%5E%7Bim%5Ctheta%7D%20%2B%20e%5E%7Bi(m-2)%5Ctheta%7D%20%2B%20%5Cdots%20%2B%20e%5E%7B-im%5Ctheta%7D)%20%5C%5C%20%5C%5C%20%26%20%3D%20e%5E%7Bi(n%2Bm)%5Ctheta%7D%20%2B%20e%5E%7Bi(n%2Bm-2)%5Ctheta%7D%20%2B%20%5Cdots%20%2B%20e%5E%7Bi(n-m)%5Ctheta%7D%20%5C%5C%26%2B%20e%5E%7Bi(n%2Bm-2)%5Ctheta%7D%20%2B%20e%5E%7Bi(n%2Bm-4)%5Ctheta%7D%20%2B%20%5Cdots%20%2B%20e%5E%7Bi(n-m-2)%5Ctheta%7D%20%5C%5C%20%26%2B%20%5Cdots%20%5C%5C%26%2Be%5E%7Bi(-n%2Bm)%5Ctheta%7D%20%2B%20e%5E%7Bi(-n%2Bm-2)%5Ctheta%7D%20%2B%20%5Cdots%20%2B%20e%5E%7Bi(-n-m)%5Ctheta%7D.%5Cend%7Baligned%7D%24%24%20Now%20look%20at%20the%20equation%20we%20get%2C%20if%20we%20think%20it%20as%20an%20%24n%2B1%24%20by%20%24m%2B1%24%20table%2C%20then%20each%20time%20we%20move%20downwards%20a%20cell%2C%20or%20rightwards%20a%20cell%2C%20the%20coefficients%20in%20the%20exponential%20function%20decreased%20by%20%242%24.%0A%0A%20%20%20%20%3Cimg%20src%3D%22public%2FPasted%20image%2020220302165337.png%22%20width%3D%22600%22%20%2F%3E%0A%0A%20%20%20%20If%20we%20look%20at%20the%20outer-most%20layer%20(colored%20by%20grey)%2C%20the%20coefficients%20go%20from%20%24n%2Bm%24%20all%20the%20way%20to%20%24-n-m%24%2C%20each%20time%20decreased%20by%20%242%24.%20Similarly%2C%20if%20we%20look%20at%20the%20second-outer-most%20layer%2C%20colored%20by%20red%2C%20the%20coefficients%20go%20from%20%24n%2Bm-2%24%20to%20%24-n-m%2B2%24%2C%20and%20so%20on.%20If%20%24n%20%3D%20m%24%20then%20this%20table%20is%20square%2C%20and%20in%20the%20end%20we%20left%20with%20bottom-left%20corner%2C%20which%20is%20%24e%5E%7Bi(-n%2Bm)%5Ctheta%7D%20%3D%20e%5E0%20%3D%201%24.%20More%20generally%20if%20%24m%5Cge%20n%24%2C%20we%20will%20be%20left%20with%20terms%20with%20coefficients%20go%20from%20%24-n%2Bm%24%20to%20%24n-m%24%20(colored%20by%20purple).%20Group%20the%20summands%20like%20that%2C%20we%20get%3A%20%24%24%5Cbegin%7Baligned%7D%5Cchi_n%5Cchi_m%20%26%3D%20e%5E%7Bi(n%2Bm)%5Ctheta%7D%2B%5Cdots%20%2B%20e%5E%7Bi(-n-m)%5Ctheta%7D%5C%5C%26%2B%20e%5E%7Bi(n%2Bm-2)%5Ctheta%7D%2B%5Cdots%20%2B%20e%5E%7Bi(-n-m%2B2)%5Ctheta%7D%5C%5C%26%2B%5Cdots%20%5C%5C%26%2Be%5E%7Bi(-n%2Bm)%5Ctheta%7D%2B%5Cdots%20%2B%20e%5E%7Bi(n-m)%5Ctheta%7D%5C%5C%26%3D%5Cchi_%7Bn%2Bm%7D%2B%5Cchi_%7Bn%2Bm-2%7D%2B%5Cdots%2B%5Cchi_%7B-n%2Bm%7D%20%5C%5C%26%3D%20%5Csum_%7Bk%3D0%7D%5En%5Cchi_%7Bn%2Bm-2k%7D.%5Cend%7Baligned%7D%24%24%20And%20we%20get%20the%20decomposition.%0A%0A%20%20%20%20Not%20sure%20about%20the%20information%20encoded%E2%80%A6%20One%20thing%20is%20that%20since%20we%20know%20if%20%24U%2C%20V%24%20are%20some%20modules%20and%20%24W%20%3D%20U%20%5Cotimes%20V%24%20is%20their%20tensor%20product%2C%20then%20%24%5Cchi_W%20%3D%20%5Cchi_U%20%5Cchi_V%24.%20From%20above%20calculation%20we%20will%20get%20that%20the%20tensor%20product%20of%20the%20modules%20correspond%20to%20%24%5Cchi_m%24%20and%20%24%5Cchi_n%24%20is%20the%20direct%20sum%20of%20the%20(simple)%20modules%20correspond%20to%20%24%5Cchi_%7Bn%2Bm%7D%2C%20%5Cchi_%7Bn%2Bm-2%7D%2C%20%5Cdots%24%2C%20and%20%24%5Cchi_%7B-n%2Bm%7D%24.%0A%20%20%20%20%22%22%22)%0A%20%20%20%20return%0A%0A%0A%40app.cell%0Adef%20_(mo)%3A%0A%20%20%20%20mo.md(r%22%22%22%0A%20%20%20%20%23%20Spring%20Semester%20Homework%204%0A%20%20%20%20%22%22%22)%0A%20%20%20%20return%0A%0A%0A%40app.cell%0Adef%20_(mo)%3A%0A%20%20%20%20mo.md(r%22%22%22%0A%20%20%20%20%23%23%20Exercise%201%0A%0A%20%20%20%20**Assuming%20%24%5Cmathbb%7BZ%7D%5B%5Csqrt%7B-19%7D%5D%24%20contains%20no%20non-principal%20ideals%20%24I%24%20such%20that%20%24I%5E5%24%20is%20principal%2C%20find%20all%20integer%20solutions%20of%20%24x%5E2%20%2B%2019%20%3D%20y%5E5%24**.%0A%0A%20%20%20%20Solution%0A%0A%20%20%20%20Write%20the%20equation%20as%20%24(x%20-%20%5Csqrt%7B-19%7D)(x%2B%5Csqrt%7B-19%7D)%20%3D%20y%5E5%24.%20First%20we%20see%20%24x%20-%20%5Csqrt%7B-19%7D%24%20and%20%24x%2B%5Csqrt%7B-19%7D%24%20are%20coprime%3A%20if%20they%20have%20a%20non-trivial%20common%20divisor%20%24z%24%2C%20then%20%24z%24%20divides%20their%20difference%20%242%5Csqrt%7B-19%7D%24%2C%20so%20%24z%24%20is%20either%20(divisor%20of)%20%242%24%20or%20%24%5Csqrt%7B-19%7D%24.%20But%20%242%20%5Cnmid%20x%5Cpm%5Csqrt%7B-19%7D%24%2C%20so%20%24z%20%3D%20%5Csqrt%7B-19%7D%24%20so%20that%20%24x%24%20is%20a%20multiple%20of%20%24%5Csqrt%7B-19%7D%24%20so%20is%20a%20multiple%20of%20%2419%24%2C%20so%20%24y%24%20is%20a%20multiple%20of%20%2419%24%2C%20thus%20%24y%5E5%20-%20x%5E2%24%20should%20be%20a%20multiple%20of%20%2419%5E2%24%20but%20%2419%24%20is%20not.%0A%0A%20%20%20%20Thus%20both%20%24(x%20-%20%5Csqrt%7B-19%7D)%24%20and%20%24(x%2B%5Csqrt%7B-19%7D)%24%20should%20be%20%24I%5E5%24%20for%20some%20%24I%24.%20By%20the%20assumption%2C%20%24I%24%20must%20be%20principal.%20So%20we%20have%20%24x%20%2B%20%5Csqrt%7B-19%7D%20%3D%20(a%20%2B%20b%20%5Csqrt%7B-19%7D)%5E5%24%20for%20some%20integers%20%24a%2C%20b%24.%20Expand%20it%20we%20get%20%24x%20%3D%20a%5E5%20-%20190a%5E3b%5E2%20%2B%201805ab%5E4%24%20and%20%241%20%3D%20b(5a%5E4-190a%5E2b%5E2%2B361b%5E4)%24.%20From%20the%20second%20equation%20we%20get%20%24b%20%3D%20%5Cpm%201%24%2C%20so%20%245a%5E4%20-%20190a%5E2%20%2B%20361%20%3D%20%5Cpm%201%24.%20We%20can%20just%20solve%20it%20as%20a%20quadratic%20equation%2C%20and%20it%20is%20not%20very%20hard%20to%20get%20%24a%5E2%20%3D%2036%24%20thus%20%24a%20%3D%20%5Cpm%206%24%20are%20the%20only%20integer%20solutions%20(in%20which%20case%20%24b%20%3D%201%24).%20Plugin%20to%20the%20first%20equation%2C%20we%20get%20%24x%20%3D%20-22434%24%20if%20%24a%20%3D%206%24%20or%20%2422434%24%20if%20%24a%20%3D%20-6%24.%20So%20%24y%5E5%20%3D%2022434%5E2%2B19%24%20and%20we%20can%20get%20%24y%20%3D%2055%24.%0A%0A%20%20%20%20So%20we%20get%20two%20solutions%2C%20%24x%20%3D%20%5Cpm%2022434%24%20and%20%24y%20%3D%2055%24.%0A%0A%20%20%20%20%23%23%20Exercise%202%0A%0A%20%20%20%20**Let%20%24R%24%20be%20a%20noetherian%20domain%20with%20the%20property%20that%20every%20prime%20ideal%20of%20%24R%24%20is%20principal.%20Prove%20that%20%24R%24%20is%20a%20PID**.%0A%0A%20%20%20%20Proof%0A%0A%20%20%20%20Let%20%24%5Cmathscr%7BR%7D%24%20be%20the%20class%20of%20all%20non-principal%20ideals%20of%20%24R%24%20and%20suppose%20it%20is%20non-empty.%20Since%20%24R%24%20is%20noetherian%2C%20%24%5Cmathscr%7BR%7D%24%20has%20a%20maximal%20element%2C%20let's%20call%20it%20%24Q%24.%20By%20construction%20%24Q%24%20is%20not%20principal%2C%20thus%20by%20assumption%20it%20is%20not%20prime.%20So%20we%20can%20find%20%24x%2C%20y%20%5Cnotin%20Q%24%20yet%20%24xy%20%5Cin%20Q%24.%20So%20%24Q'%20%3D%20Q%20%2B%20(x)%24%20is%20larger%20than%20%24Q%24%20thus%20principal.%20Also%20consider%20the%20ideal%20quotient%20%24Q%3A(x)%24%20%24%3D%5Clbrace%20r%20%5Cin%20R%20%7C%20r(x)%20%5Csubseteq%20Q%20%5Crbrace%24%2C%20we%20know%20it%20is%20an%20ideal%20and%20apparently%20it%20contains%20%24Q%24%2C%20also%20we%20should%20see%20%24y%20%5Cin%20Q%3A(x)%24%20because%20%24xy%20%5Cin%20Q%24%2C%20so%20%24Q%3A(x)%24%20is%20also%20larger%20than%20%24Q%24%20thus%20principal.%20Say%20%24Q'%20%3D%20(a)%24%20and%20%24Q%3A(x)%20%3D%20(b)%24.%20Consider%20the%20principal%20ideal%20%24(ab)%24%3A%0A%0A%20%20%20%201.%20%24(ab)%20%5Csubseteq%20Q%24%3A%20By%20construction%20%24(a)%20%3D%20Q%2B(x)%24%20so%20%24a%24%20can%20be%20written%20as%20%24q'%20%2B%20x'%24%20for%20some%20%24q'%20%5Cin%20Q%24%20and%20%24x'%20%5Cin%20(x)%24%2C%20so%20%24ab%20%3D%20q'b%2Bx'b%24%2C%20where%20%24q'b%20%5Cin%20Q%24%20because%20%24Q%24%20is%20an%20ideal%2C%20and%20%24x'b%20%5Cin%20Q%24%20by%20definition%20of%20ideal%20quotient%20(by%20saying%20domain%20we%20assume%20%24R%24%20is%20commutative%20ring)%2C%20thus%20%24ab%20%5Cin%20Q%24%20so%20%24(ab)%20%5Csubseteq%20Q%24%3B%0A%20%20%20%202.%20%24Q%20%5Csubseteq%20(ab)%24%3A%20Let%20%24q%20%5Cin%20Q%24%2C%20by%20above%20we%20know%20%24Q%20%5Csubset%20(a)%24%20so%20%24q%20%3D%20ra%24%20%24%3Dr%20(q'%2Bx')%24%20%24%3D%20rq'%20%2B%20rx'%24.%20Since%20%24rq'%20%5Cin%20Q%24%2C%20we%20have%20%24rx'%20%3D%20q-rq'%20%5Cin%20Q%24%2C%20thus%20%24r(x)%20%5Csubseteq%20Q%24%2C%20which%20by%20definition%20means%20%24r%20%5Cin%20Q%3A(x)%20%3D%20(b)%24%2C%20so%20%24r%20%3D%20r'b%24%2C%20so%20%24q%20%3D%20r'ba%20%5Cin%20(ab)%24.%0A%0A%20%20%20%20So%20%24Q%24%20is%20principal%20and%20we%20have%20a%20contradiction%2C%20thus%20there%20is%20no%20non-principal%20ideal%2C%20i.e.%20%24R%24%20is%20PID.%0A%0A%20%20%20%20%23%23%20Exercise%203%0A%0A%20%20%20%20**Let%20%24R%20%3D%20%5Cmathbb%7BZ%7D%5B%5Csqrt%7B-5%7D%5D%24%2C%20and%20%24P%20%3D%20(2%2C%201%20%2B%20%5Csqrt%7B-5%7D)%24%20%24%3D(2%2C1-%5Csqrt%7B-5%7D)%24%2C%20%24Q_1%20%3D%20(3%2C%201%2B%5Csqrt%7B-5%7D)%24%2C%20%24Q_2%20%3D%20(3%2C%201-%5Csqrt%7B-5%7D)%24%20be%20ideals%20of%20%24R%24**.%0A%0A%20%20%20%201.%20**Check%20that%20%24P%2CQ_1%2C%20Q_2%24%20are%20maximal%20ideals%20in%20%24R%24**%3B%0A%20%20%20%202.%20**Verify%20that%20%24P%5E2%2C%20PQ_1%2C%20PQ_2%24%20and%20%24Q_1Q_2%24%20are%20principal**%3B%0A%20%20%20%203.%20**Check%20that%20the%20factorizations%20%246%20%3D%202%20%5Ccdot%203%20%3D%20(1%2B%5Csqrt%7B-5%7D)(1-%5Csqrt%7B-5%7D)%24%20lead%20to%20the%20same%20factorization%20of%20%24I%20%3D%20(6)%24%20into%20prime%20ideals%20in%20%24R%24**.%0A%0A%20%20%20%20Proof%0A%20%20%20%201.%20We%20know%20%24%5Cmathbb%7BZ%7D%5B%5Csqrt%7B-5%7D%5D%20%3D%20%5Clbrace%20a%2Bb%5Csqrt%7B-5%7D%7C%20a%2C%20b%20%5Cin%20%5Cmathbb%7BZ%7D%20%5Crbrace%24.%20Consider%20a%20map%20%24%5Cvarphi%3A%20%5Cmathbb%7BZ%7D%5B%5Csqrt%7B-5%7D%5D%20%5Cto%20%5Cmathbb%7BZ%7D_2%24%20given%20by%20%24a%2Bb%5Csqrt%7B-5%7D%20%5Cmapsto%20a%2Bb%20%5Cpmod%7B2%7D%24.%0A%20%20%20%20%20%20%20%201.%20It%20is%20a%20homomorphism%3A%20Let%20%24a%2Bb%5Csqrt%7B-5%7D%24%20and%20%24c%2Bd%5Csqrt%7B-5%7D%24%20be%20arbitrary%20elements%20of%20%24%5Cmathbb%7BZ%7D%5B%5Csqrt%7B-5%7D%5D%24%2C%20then%20%24%5Cvarphi((a%2Bb%5Csqrt%7B-5%7D)(c%2Bd%5Csqrt%7B-5%7D))%24%20%24%3D%5Cvarphi(ac-5bd%20%2B%20(bc%2Bad)%5Csqrt%7B-5%7D)%24%20%24%3Dac-5bd%2Bbc%2Bad%24%20%24%3D%20ac%2Bbd%2Bbc%2Bad%20%5Cpmod%7B2%7D%24%3B%20and%20%24%5Cvarphi(a%2Bb%5Csqrt%7B-5%7D)%5Cvarphi(c%2Bd%5Csqrt%7B-5%7D)%24%20%24%3D(a%2Bb)(c%2Bd)%20%3D%24%20%24ac%2Bad%2Bbc%2Bbd%24.%20So%20they%20are%20the%20same%2C%20%24%5Cvarphi%24%20is%20a%20homomorphism%3B%0A%20%20%20%20%20%20%20%202.%20The%20kernel%20is%20%24P%24%3A%20%24%5Cvarphi(2)%20%3D%20%5Cvarphi(1-%5Csqrt%7B-5%7D)%20%3D%200%24%20so%20%24P%20%5Csubseteq%20%5Cker%7B%5Cvarphi%7D%24.%20If%20%24a%2Bb%5Csqrt%7B-5%7D%20%5Cin%20%5Cker%7B%5Cvarphi%7D%24%20then%20%24a%2Bb%20%5Cequiv%200%20%5Cpmod%7B2%7D%24%20%24%5Cimplies%20a%20%3D%20-b%2B2k%24%2C%20so%20%24a%2Bb%5Csqrt%7B-5%7D%24%20%24%3D%20-b%2B2k%2Bb%5Csqrt%7B-5%7D%24%20%24%3D2k%20-%20b(1-%5Csqrt%7B-5%7D)%24%20%24%5Cin%20P%24.%0A%20%20%20%20%20%20%20%20So%20%24R%2FP%20%5Ccong%20%5Cmathbb%7BZ%7D_2%24%20is%20a%20field%2C%20%24P%24%20is%20maximal.%0A%20%20%20%20%20%20%20%20Similarly%2C%20the%20isomorphism%20%24%5Cmathbb%7BZ%7D%5B%5Csqrt%7B-5%7D%5D%20%5Cto%20%5Cmathbb%7BZ%7D_3%24%20given%20by%20%24a%2Bb%5Csqrt%7B-5%7D%20%5Cmapsto%20a%2Bb%24%20proves%20%24Q_2%24%20is%20maximal%2C%20and%20the%20isomorphism%20%24%5Cmathbb%7BZ%7D%5B%5Csqrt%7B-5%7D%5D%20%5Cto%20%5Cmathbb%7BZ%7D_3%24%20given%20by%20%24a%2Bb%5Csqrt%7B-5%7D%20%5Cmapsto%20a-b%24%20proves%20%24Q_1%24%20is%20maximal%3B%0A%20%20%20%202.%20A%20generic%20element%20of%20%24P%24%20is%20%242a%2Bb(1%2B%5Csqrt%7B-5%7D)%24%2C%20which%20can%20also%20be%20written%20as%20%242c%2Bd(1-%5Csqrt%7B-5%7D)%24%2C%20so%20%24%24P%5E2%20%3D%20%5Clbrace%20%5Csum_%7Bi%3D1%7D%5En%20(2a_i%2Bb_i(1%2B%5Csqrt%7B-5%7D)(2c_i%2Bd_i(1-%5Csqrt%7B-5%7D))%20%5Crbrace.%24%24%20Expand%20and%20we%20can%20see%20each%20summand%20is%20of%20the%20form%20%244a_ic_i%20%2B%202c_ib_i(1%2B%5Csqrt%7B-5%7D)%20%2B%202a_i%20d_i(1-%5Csqrt%7B-5%7D)%20%2B%206b_i%20d_i%24.%20In%20particular%20each%20term%20is%20a%20multiple%20of%20%242%24%2C%20so%20%24P%5E2%20%5Csubseteq%20(2)%24.%20Of%20course%20%24(2)%20%5Csubseteq%20P%5E2%24%20so%20we%20have%20%24P%5E2%20%3D%20(2)%24%20is%20principal.%0A%20%20%20%20%20%20%20%20For%20%24PQ_1%24%20and%20%24PQ_2%24%2C%20we%20should%20see%20%24P%20%2B%20Q_1%24%20and%20%24P%2BQ_2%24%20are%20both%20the%20entire%20%24R%24%20because%20%24-2%2B3%20%3D%201%24.%20So%20Chinese%20Remainder%20Theorem%20says%20%24PQ_1%20%3D%20P%20%5Ccap%20Q_1%24%20%24%3D(1%2B%5Csqrt%7B-5%7D)%24.%20To%20see%20that%3A%20of%20course%20%24(1%2B%5Csqrt%7B-5%7D)%20%5Csubseteq%20P%20%5Ccap%20Q_1%24%2C%20if%20%24x%20%5Cin%20P%20%5Ccap%20Q_1%24%20then%20%24x%24%20%24%3D%202a%20%2B%20b(1%2B%5Csqrt%7B-5%7D)%24%20%24%3D%203c%20%2B%20d(1%2B%5Csqrt%7B-5%7D)%24%20so%20that%20%242a%20%2B%20(b-d)(1%2B%5Csqrt%7B-5%7D)%20%3D%203c%24%2C%20since%20%243%24%20is%20irreducible%20and%20%24%5Cgcd%7B(2%2C3)%7D%20%3D%201%24%2C%20we%20need%20%24a%24%20to%20be%20a%20multiple%20of%20%243%24%20thus%20%242a%24%20is%20a%20multiple%20of%20%246%20%3D%20(1%2B%5Csqrt%7B-5%7D)(1-%5Csqrt%7B-5%7D)%24%2C%20so%20that%20%242a%20%5Cin%20(1%2B%5Csqrt%7B-5%7D)%24%2C%20thus%20%24x%20%5Cin%20(1%2B%5Csqrt%7B-5%7D)%24.%20Similarly%20%24PQ_2%20%3D%20P%20%5Ccap%20Q_2%24%20%24%3D%20(1-%5Csqrt%7B-5%7D)%24.%0A%20%20%20%20%20%20%20%20%24Q_1%20%2B%20Q_2%20%3D%20R%24%20as%20well%2C%20because%20%24-(1%2B%5Csqrt%7B-5%7D%20%2B%201%20-%20%5Csqrt%7B-5%7D)%2B3%20%3D%201%24.%20So%20%24Q_1Q_2%20%3D%20Q_1%20%5Ccap%20Q_2%24%20%24%3D%20(3)%24%3B%0A%20%20%20%203.%20Maximal%20ideals%20are%20primes%2C%20so%20by%20part%201%20we%20know%20%24P%2C%20Q_1%24%20and%20%24Q_2%24%20are%20prime%20ideals.%20Now%20use%20part%202%2C%20%24(6)%20%3D%20(2)(3)%24%20%24%3D%20(P%5E2)(Q_1Q_2)%24%20is%20a%20factorization%3B%20and%20the%20other%20one%3A%20%24(6)%20%3D%20(1%2B%5Csqrt%7B-5%7D)(1-%5Csqrt%7B-5%7D)%24%20%24%3D%20(PQ_1)(PQ_2)%24.%20Since%20%24R%24%20is%20commutative%2C%20order%20of%20product%20of%20ideals%20does%20not%20matter%2C%20so%20the%20factorizations%20are%20the%20same.%0A%0A%20%20%20%20%23%23%20Exercise%204%0A%0A%20%20%20%20**Let%20%24R%24%20be%20a%20domain%2C%20%24I%24%20and%20%24J%24%20be%20fractional%20%24R%24-ideals%20and%20%24P%24%20be%20a%20prime%20in%20%24R%24.%20Verify%20that%20%24I_P%20%2B%20J_P%20%3D%20(I%2BJ)_P%24%20and%20%24I_PJ_P%20%3D%20(IJ)_P%24**.%0A%0A%20%20%20%20Proof%0A%0A%20%20%20%20By%20definition%2C%20%24I_P%20%3D%20%5Clbrace%20a%2Fb%20%7C%20a%20%5Cin%20I%2C%20b%20%5Cin%20R-P%20%5Crbrace%24%20and%20%24J_P%20%3D%20%5Clbrace%20c%2Fd%20%7C%20c%20%5Cin%20J%2C%20d%20%5Cin%20R-P%20%5Crbrace%24%2C%20so%20%24I_P%2BJ_P%20%3D%20%5Clbrace%20%5Cfrac%7Ba%7D%7Bb%7D%20%2B%20%5Cfrac%7Bc%7D%7Bd%7D%20%3D%20%5Cfrac%7Bad%2Bcb%7D%7Bbd%7D%20%7C%20a%20%5Cin%20I%2C%20c%20%5Cin%20J%2C%20b%2Cd%20%5Cin%20R-P%20%5Crbrace%24.%20Consider%20a%20generic%20term%20%24%5Cfrac%7Bad%2Bcb%7D%7Bbd%7D%24.%20Since%20%24P%24%20is%20prime%2C%20%24b%20%2Cd%20%5Cnotin%20P%20%5Cimplies%20bd%20%5Cnotin%20P%24%3B%20%24ad%2Bcb%24%20lies%20in%20%24I%2BJ%24%20because%20%24a%20%5Cin%20I%24%20and%20%24c%20%5Cin%20J%24.%20So%20%24%5Cfrac%7Bad%2Bcb%7D%7Bbd%7D%20%5Cin%20(I%2BJ)_P%24%2C%20meaning%20%24I_P%2BJ_P%20%5Csubseteq%20(I%2BJ)_P%24.%0A%0A%20%20%20%20%24(I%2BJ)_P%20%3D%20%5Clbrace%20%5Cfrac%7Ba%2Bc%7D%7Bb%7D%20%7C%20a%5Cin%20I%2C%20c%20%5Cin%20J%2C%20b%5Cin%20R-P%20%5Crbrace%24%2C%20clearly%20we%20may%20write%20any%20of%20its%20element%20as%20some%20%24%5Cfrac%7Ba%7D%7Bb%7D%20%2B%20%5Cfrac%7Bc%7D%7Bb%7D%24%20and%20where%20%24%5Cfrac%7Ba%7D%7Bb%7D%24%20lies%20in%20%24I_P%24%20and%20%24%5Cfrac%7Bc%7D%7Bd%7D%24%20lies%20in%20%24J_P%24%2C%20so%20we%20have%20the%20other%20inclusion.%0A%0A%20%20%20%20%24I_PJ_P%20%3D%20%5Clbrace%20%5Cfrac%7Ba_1%7D%7Bb_1%7D%5Cfrac%7Bc_1%7D%7Bd_1%7D%20%2B%20%5Cdots%20%2B%20%5Cfrac%7Ba_n%7D%7Bb_n%7D%5Cfrac%7Bc_n%7D%7Bd_n%7D%7C%20a_i%20%5Cin%20I%2C%20c_i%20%5Cin%20J%2C%20b_i%2C%20d_i%20%5Cin%20R-P%5Crbrace%24.%20Since%20%24b_1%2Cd_1%20%5Cnotin%20P%24%2C%20a%20prime%20ideal%2C%20%24b_1d_1%5Cnotin%20P%24%2C%20thus%20finitely%20product%20%24b_1d_1%5Cdots%20b_nd_n%20%5Cnotin%20P%24%2C%20combine%20all%20fractions%20and%20we%20can%20see%20the%20numerator%20is%20in%20%24IJ%24.%20So%20%24I_PJ_P%20%5Csubseteq%20(IJ)_P%24.%0A%0A%20%20%20%20%24(IJ)_P%20%3D%20%5Clbrace%20%5Cfrac%7Ba_1c_1%20%2B%20%5Cdots%20%2B%20a_nc_n%7D%7Bb%7D%20%7C%20a_i%5Cin%20I%2C%20c_i%20%5Cin%20J%2C%20b%20%5Cin%20R-P%5Crbrace%24%2C%20split%20the%20fraction%20to%20%24%5Csum%5Cfrac%7Ba_ic_i%7D%7Bb%7D%24%20where%20each%20term%20can%20be%20written%20as%20%24%5Cfrac%7Ba_i%7D%7Bb%7D%20%5Cfrac%7Bc_i%7D%7B1%7D%24.%20Since%20prime%20ideal%20is%20by%20definition%20proper%2C%20%241%20%5Cnotin%20P%24%2C%20so%20%24%5Cfrac%7Ba_i%7D%7Bb%7D%5Cin%20I_P%24%20and%20%24%5Cfrac%7Bc_i%7D%7B1%7D%20%5Cin%20J_P%24%2C%20thus%20%24(IJ)_P%20%5Csubseteq%20I_PJ_P%24.%0A%0A%20%20%20%20%23%23%20Exercise%205%0A%0A%20%20%20%20**Let%20%24F%24%20be%20a%20field%20and%20%24R%20%3D%20F%5B%5Bt%5D%5D%24%20be%20the%20ring%20of%20formal%20power%20series%20over%20%24F%24.%20Show%20that%20%24R%24%20is%20a%20discrete%20valuation%20ring**.%0A%0A%20%20%20%20Proof%0A%0A%20%20%20%20Claim%3A%20%24f%20%3D%20%5Csum%5Climits_%7Bn%20%3D%200%7D%5E%7B%5Cinfty%7Da_nt%5En%24%20%24%5Cin%20R%24%20is%20a%20unit%20if%20and%20only%20if%20%24a_0%20%5Cne%200%24.%0A%0A%20%20%20%20(%24%5Cimplies%24)%20Suppose%20%24f%24%20is%20a%20unit%2C%20then%20we%20must%20have%20some%20%24g%20%3D%20%5Csum%20b_nt%5En%24%20so%20that%20%24fg%20%3D%201%24%2C%20look%20at%20the%20constant%20term%2C%20we%20must%20have%20%24a_0b_0%20%3D%201%24%20so%20in%20particular%20%24a_0%24%20cannot%20be%20%240%24.%0A%0A%20%20%20%20(%24%5Cimpliedby%24)%20Suppose%20%24f%20%3D%20%5Csum%5Climits_%7Bn%20%3D%200%7D%5E%7B%5Cinfty%7Da_nt%5En%24%20with%20%24a_0%20%5Cne%200%24%2C%20take%20%24g%20%3D%20%5Csum%5Climits_%7Bn%3D0%7D%5E%7B%5Cinfty%7Db_nt%5En%24%20such%20that%20%24b_0%20%3D%20%5Cfrac%7B1%7D%7Ba_0%7D%24%2C%20%24b_1%20%3D%20%5Cfrac%7B-a_1b_0%7D%7Ba_0%7D%24%2C%20%24%5Cdots%24%2C%20%24b_n%20%3D%20%5Cfrac%7B%5Csum%5Climits_%7Bi%3D1%7D%5En-a_ib_%7Bn-i%7D%7D%7Ba_0%7D%24%2C%20so%20that%20the%20coefficient%20of%20the%20constant%20term%20is%20%241%24%2C%20and%20coefficient%20of%20%24x%5En%24%20is%20%240%24%20for%20any%20%24n%20%5Cge%201%24%2C%20thus%20%24fg%20%3D%201%24%20and%20%24f%24%20is%20a%20unit.%0A%0A%20%20%20%20In%20particular%2C%20%24R%20-%20R%5E%7B*%7D%24%20%24%3D(t)%24%20is%20the%20only%20maximal%20ideal%2C%20%24R%24%20is%20local.%20Any%20ideal%20in%20%24R%24%20can%20be%20expressed%20in%20the%20form%20%24(t%5En)%2C%20n%20%5Cge%201%24%20because%20%24f%20%5Cin%20R%24%20are%20infinite%20series%20-%20just%20take%20the%20smallest%20power%20among%20all%20terms%20of%20the%20generators%20(for%20example%2C%20%24(2x%5E3%20%2B%206x%5E8%2C%20x%5E5)%24%20is%20just%20%24(x%5E3)%24).%20Thus%20%24R%24%20is%20a%20PID.%20By%20definition%2C%20a%20local%20PID%20is%20a%20DVR.%0A%0A%20%20%20%20%23%23%20Exercise%206%0A%0A%20%20%20%20**Let%20%24R%20%3D%20%5Cmathbb%7BZ%7D%5B%5Csqrt%7B-61%7D%5D%24%20and%20%24I%20%3D%20(2%2C%203%20%2B%20%5Csqrt%7B-61%7D)%24%2C%20%24J%20%3D%20(5%2C%203%2B%20%5Csqrt%7B-61%7D)%24**.%0A%0A%20%20%20%201.%20**For%20each%20prime%20%24P%24%20in%20%24R%24%2C%20find%20the%20local%20generators%20for%20%24I_P%24%20and%20%24J_P%24**%3B%0A%20%20%20%202.**Conclude%20that%20%24I%24%20and%20%24J%24%20are%20invertible%20ideals%2C%20and%20determine%20the%20order%20of%20their%20classes%20in%20%24Pic(R)%24**.%0A%0A%20%20%20%20Proof%0A%0A%20%20%20%201.%20We%20can%20see%20that%20%24R%2FI%20%5Ccong%20%5Cmathbb%7BZ%7D_2%24%20and%20%24R%2FJ%20%5Ccong%20%5Cmathbb%7BZ%7D_5%24%2C%20so%20%24I%2C%20J%24%20are%20both%20maximal%20(and%20prime).%0A%20%20%20%20%20%20%20%201.%20If%20%24P%20%3D%20I%24%2C%20since%20%24%5Cfrac%7B2%7D%7B3%2B%5Csqrt%7B-61%7D%7D%20%3D%20%5Cfrac%7B3-%5Csqrt%7B-61%7D%7D%7B35%7D%24%20and%20the%20denominator%20is%20not%20in%20%24I%24%2C%20%24%5Cfrac%7B2%7D%7B3%2B%5Csqrt%7B-61%7D%7D%20%5Cin%20R_I%24.%20In%20particular%2C%20it%20means%20%242%24%20is%20a%20multiple%20of%20%243%2B%5Csqrt%7B-61%7D%24%20in%20the%20localization%20thus%20is%20redundant%20as%20a%20generator%2C%20we%20may%20write%20%24I_I%20%3D%20(3%2B%5Csqrt%7B-61%7D)%24%20and%20it%20is%20principal%3B%0A%20%20%20%20%20%20%20%202.%20If%20%24P%20%5Cne%20I%24%2C%20then%20there%20is%20some%20non-zero%20%24c%20%5Cin%20I%24%20such%20that%20%24c%20%5Cnotin%20P%24%2C%20then%20any%20element%20in%20%24R_P%24%2C%20having%20the%20form%20%24%5Cfrac%7Ba%7D%7Bb%7D%2C%20a%20%5Cin%20R%2C%20b%20%5Cin%20R-P%24%20can%20be%20written%20as%20%24%5Cfrac%7Bac%7D%7Bbc%7D%24%20with%20now%20%24ac%20%5Cin%20I%24%20and%20%24bc%20%5Cin%20R-P%24%20because%20%24P%24%20is%20prime%2C%20so%20that%20we%20see%20%24R_P%20%5Csubseteq%20I_P%24.%20Since%20%24I_P%20%5Csubseteq%20R_P%24%20is%20immediate%20from%20definition%2C%20we%20see%20now%20%24I_P%20%3D%20R_P%24%20is%20the%20entire%20ring%2C%20and%20is%20of%20course%20generated%20by%20%241%24%3B%0A%20%20%20%20%20%20%20%203.%20If%20%24P%20%3D%20J%24%2C%20similarly%20%24%5Cfrac%7B5%7D%7B3%2B%5Csqrt%7B-61%7D%7D%20%3D%20%5Cfrac%7B3-%5Csqrt%7B-61%7D%7D%7B14%7D%24%20and%20thus%20%24J_J%20%3D%20(5)%24%3B%0A%20%20%20%20%20%20%20%204.%20If%20%24P%20%5Cne%20J%24%2C%20similarly%20%24J_P%20%3D%20R_P%24%20can%20be%20generated%20by%20%241%24%3B%0A%20%20%20%202.%20By%20part%201%2C%20%24I_P%24%20and%20%24J_P%24%20are%20principal%20in%20%24R_P%24%20for%20all%20prime%20ideals%20%24P%20%5Csubset%20R%24%2C%20and%20by%20assumption%20%24I%24%20and%20%24J%24%20are%20finitely%20generated.%20By%20the%20theorem%20we%20learnt%20in%20class%2C%20we%20see%20%24I%24%20and%20%24J%24%20are%20invertible.%0A%20%20%20%20%20%20%20%20Consider%20%24I%5E2%3D%20(4%2C%20-52%2B6%5Csqrt%7B-61%7D%2C%206%2B2%5Csqrt%7B-61%7D)%24%2C%20we%20can%20see%20%24I%5E2%20%3D%20(2)%24%2C%20thus%20%24I%5E2%20%5Cin%20%5Cmathcal%7BP%7D(R)%24%2C%20and%20thus%20%24I%24%20has%20order%20%242%24%20in%20the%20Picard%20group.%0A%20%20%20%20%20%20%20%20For%20%24J%24%20it%20is%20a%20little%20more%20complex.%20Consider%20the%20ideals%20%24M%20%3D%20(8%2B%5Csqrt%7B-61%7D)%24%20and%20%24N%20%3D%20(8-%5Csqrt%7B-61%7D)%24%2C%20it%20is%20not%20hard%20too%20see%20%24M%2BN%20%3D%20R%24%20since%20both%20%2416%24%20and%20%24125%24%20are%20in%20%24M%2BN%24%20thus%20%241%20%5Cin%20M%2BN%24.%20Also%20%24MN%20%3D%20(125)%20%3D%20(5)%5E3%24%2C%20we%20conclude%20using%20a%20lemma%20learnt%20in%20class%20that%20%24(8%2B%5Csqrt%7B-61%7D)%20%3D%20(5%2C%208%2B%5Csqrt%7B-61%7D)%5E3%20%3D%20(5%2C3%2B%5Csqrt%7B-61%7D)%5E3%24%2C%20and%20thus%20%24J%5E3%20%5Cin%20%5Cmathcal%7BP%7D(R)%24%20and%20thus%20%24J%24%20has%20order%20%243%24%20in%20the%20Picard%20group.%0A%0A%20%20%20%20%23%23%20Exercise%207%0A%0A%20%20%20%20**Let%20%24%5Calpha%24%20be%20a%20complex%20root%20of%20%24f(X)%20%3D%20X%5E3%20-%20X%20-%201%24%20and%20%24R%20%3D%20%5Cmathbb%7BZ%7D%5B%5Calpha%5D%24**.%0A%0A%20%20%20%201.%20**Write%20down%20all%20prime%20ideals%20of%20norm%20%24%5Cle%2025%24%20in%20%24R%24**%3B%0A%20%20%20%202.%20**Verify%20that%20the%20ideals%20of%20%24R%24%20lying%20over%20%242%2C3%2C5%2C7%24%20are%20principal**%3B%0A%20%20%20%203.%20**Verify%20that%20the%20remaining%20ideals%20from%20part%201%20are%20principal%20as%20well.%20You%20may%20use%20Sage%20to%20find%20the%20generators%2C%20e.g.%20%60K.%3CALPHA%3E%20%3D%20NumberField(x%5E3-x-1)%3BIdeal(23%2CALPHA-10)%60**.%0A%0A%20%20%20%20Solution%0A%20%20%20%201.%20Since%20%24f%24%20has%20no%20roots%20in%20%24%5Cmathbb%7BZ%7D%24%2C%20it%20is%20irreducible%2C%20thus%20by%20Kummer-Dedekind%20Theorem%2C%20all%20prime%20ideals%20are%20of%20the%20form%20%24(p%2C%20g(%5Calpha))%24%20where%20%24g%24%20is%20a%20lift%20of%20a%20factor%20of%20%24f%24%20in%20%24%5Cmathbb%7BF%7D_p%5Bx%5D%24.%20So%20let's%20look%20into%20%24f%24%20per%20%24p%24%3A%0A%20%20%20%20%20%20%20%201.%20If%20%24p%20%3D%202%24%2C%20%24%5Coverline%7Bf%7D%20%3D%20x%5E3-x-1%24%2C%20so%20apparently%20%24g(%5Calpha)%20%3D%200%24%2C%20%24P%20%3D%20(2)%24%20is%20an%20ideal%20of%20norm%20%24%7C%20R%2FP%20%7C%20%3D%202%5E%7B%5Cdeg%7Bf%7D%7D%20%3D%208%24%3B%0A%20%20%20%20%20%20%20%202.%20If%20%24p%20%3D%203%24%2C%20we%20similarly%20get%20%24P%20%3D%20(3)%24%20and%20it%20has%20norm%20%2427%24%2C%20not%20what%20we%20want%3B%0A%20%20%20%20%20%20%20%203.%20If%20%24p%20%3D%205%24%2C%20%24%5Coverline%7Bf%7D%20%3D%20(x%2B3)(x%5E2%2B2x%2B3)%24%2C%20so%20we%20get%20two%20ideals%20%24(5%2C%20%5Calpha%2B3)%24%20and%20%24(5%2C%20%5Calpha%5E2%2B2x%2B3)%24%2C%20they%20are%20of%20norm%20%245%24%20and%20%2425%24%20respectively%2C%20because%20for%20example%20%24R%2F(5%2C%5Calpha%2B3)%20%3D%20%5Clbrace%20%5Coverline%7B0%7D%2C%5Coverline%7B1%7D%2C%5Coverline%7B2%7D%2C%5Coverline%7B3%7D%2C%5Coverline%7B4%7D%5Crbrace%24%3B%0A%20%20%20%20%20%20%20%204.%20If%20%24p%20%3D%207%24%2C%20%24%5Coverline%7Bf%7D%20%3D%20(x%2B2)(x%5E2%2B5x%2B3)%24%2C%20similarly%20we%20get%20%24(7%2C%20%5Calpha%2B2)%24%20of%20norm%20%247%24%20(and%20%24(7%2C%20%5Calpha%5E2%2B5%5Calpha%2B3)%24%20of%20norm%20%2449%24%2C%20which%20we%20don't%20want)%3B%0A%20%20%20%20%20%20%20%205.%20If%20%24p%20%3D%2011%24%2C%20%24%5Coverline%7Bf%7D%20%3D%20(x%2B5)(x%5E2%2B6x%2B2)%24%2C%20similarly%20we%20get%20%24(11%2C%5Calpha%2B5)%24%20of%20norm%20%2411%24%3B%0A%20%20%20%20%20%20%20%206.%20If%20%24p%20%3D%2013%24%2C%20%24%5Coverline%7Bf%7D%20%3D%20x%5E3-x-1%24%20is%20irreducible%2C%20there%20is%20no%20prime%20ideal%20of%20norm%20%24%5Cle%2025%24%20lying%20over%20%2413%24%3B%0A%20%20%20%20%20%20%20%207.%20If%20%24p%20%3D%2017%24%2C%20%24%5Coverline%7Bf%7D%20%3D%20(x%2B12)(x%5E2%2B5x%2B7)%24%2C%20we%20have%20%24(17%2C%20%5Calpha%2B12)%24%20of%20norm%20%2417%24%3B%0A%20%20%20%20%20%20%20%208.%20If%20%24p%20%3D%2019%24%2C%20%24%5Coverline%7Bf%7D%20%3D%20(x%2B13)(x%5E2%2B6x%2B16)%24%2C%20we%20have%20%24(19%2C%20%5Calpha%20%2B%2013)%24%20of%20norm%20%2419%24%3B%0A%20%20%20%20%20%20%20%209.%20And%20finally%2C%20if%20%24p%20%3D%2023%24%2C%20%24%5Coverline%7Bf%7D%20%3D%20(x%2B20)(x%2B13)%5E2%24%2C%20we%20have%20two%20ideals%20%24(23%2C%20%5Calpha%20%2B%2020)%24%20and%20%24(23%2C%20%5Calpha%2B13)%24%20of%20norm%20%2423%24%3B%20For%20any%20larger%20prime%20%24p%24%2C%20the%20norm%20will%20be%20of%20course%20larger%20than%20%2425%24%2C%20so%20we%20have%20finished%3B%0A%20%20%20%202.%20For%20%24p%20%3D%202%24%2C%20the%20corresponding%20prime%20ideal%20of%20course%20principal%20from%20above.%0A%20%20%20%20%20%20%20%20If%20%24p%20%3D%205%24%20we%20get%20an%20ideal%20%24(5%2C%5Calpha%2B3)%24%20%24%3D(5%2C%5Calpha-2)%24%2C%20notice%20that%20%24x%5E3-x-1%20%3D%20(x%5E2%2B2x%2B3)(x-2)%20%2B%205%24%2C%20but%20%24%5Calpha%5E3-%5Calpha-1%20%3D%200%24%2C%20so%20%245%20%3D%20(-%5Calpha%5E2-2%5Calpha-3)(%5Calpha-2)%24%2C%20thus%20%245%24%20is%20redundant%20as%20a%20generator%2C%20this%20is%20a%20principal%20ideal.%20This%20expression%20also%20shows%20that%20%24(5%2C%20%5Calpha%5E2%2B2%5Calpha%2B3)%20%3D%20(%5Calpha%5E2%2B2%5Calpha%2B3)%24%2C%20so%20it%20is%20principal%20as%20well.%0A%20%20%20%20%20%20%20%20If%20%24p%20%3D%207%24%20we%20get%20prime%20ideals%20%24(7%2C%5Calpha%2B2)%24%20and%20%24(7%2C%20%5Calpha%5E2-2%5Calpha%2B3)%24%2C%20we%20can%20see%20that%20they%20are%20principal%20by%20the%20expression%20%24x%5E3-x-1%20%3D%20(x%5E2-2x%2B3)(x%2B2)-7%24%2C%20in%20a%20similar%20manner%3B%0A%20%20%20%203.%20Check%20with%20Sage%2C%20they%20are%20indeed%20all%20principal%2C%20and%20we%20finished%3A%0A%20%20%20%20%20%20%20%20%60%60%60python%0A%20%20%20%20%20%20%20%20K.%20%3D%20NumberField(x%5E3-x-1)%3B%0A%20%20%20%20%20%20%20%20I%20%3D%20Ideal(11%2C%20A%2B5)%3Bprint(I)%3B%0A%20%20%20%20%20%20%20%20I%20%3D%20Ideal(17%2C%20A%2B12)%3Bprint(I)%3B%0A%20%20%20%20%20%20%20%20I%20%3D%20Ideal(19%2C%20A%2B13)%3Bprint(I)%3B%0A%20%20%20%20%20%20%20%20I%20%3D%20Ideal(23%2C%20A%2B20)%3Bprint(I)%3B%0A%20%20%20%20%20%20%20%20I%20%3D%20Ideal(23%2C%20A%2B13)%3Bprint(I)%3B%0A%20%20%20%20%20%20%20%20%3E%3E%20Fractional%20ideal%20(-2*A%5E2%20%2B%20A)%0A%20%20%20%20%20%20%20%20%3E%3E%20Fractional%20ideal%20(-A%5E2%20%2B%20A%20%2B%203)%0A%20%20%20%20%20%20%20%20%3E%3E%20Fractional%20ideal%20(2*A%5E2%20%2B%20A%20-%202)%0A%20%20%20%20%20%20%20%20%3E%3E%20Fractional%20ideal%20(-3*A%5E2%20%2B%20A%20%2B%201)%0A%20%20%20%20%20%20%20%20%3E%3E%20Fractional%20ideal%20(A%5E2%20-%203*A%20-%201)%0A%20%20%20%20%20%20%20%20%60%60%60%0A%0A%20%20%20%20%23%23%20Exercise%208%0A%0A%20%20%20%20**Determine%20which%20of%20the%20following%20rings%20are%20Dedekind%20domains**%3A%0A%0A%20%20%20%201.%20**%24%5Cmathbb%7BZ%7D%5B%5Csqrt%5B3%5D%7B2%7D%5D%24**%2C%0A%20%20%20%202.%20**%24%5Cmathbb%7BZ%7D%5B%5Csqrt%5B5%5D%7B7%7D%5D%24**%2C%0A%20%20%20%203.%20**%24%5Cmathbb%7BZ%7D%5B%5Calpha%5D%24%20where%20%24%5Calpha%5E3%20%3D%20%5Calpha%2B1%24**.%0A%0A%20%20%20%20Solution%0A%0A%20%20%20%201.%20Consider%20%24f%20%3D%20x%5E3-2%24.%20If%20%24p%20%3D%202%24%20then%20%24%5Coverline%7Bf%7D%20%3D%20x%5E3%24%20so%20it%20is%20%24%5Coverline%7Bg_1%7D%5E3%24%20if%20%24%5Coverline%7Bg_1%7D%20%3D%20x%24%2C%20remainder%20%24r_1%20%3D%20-2%20%5Cnmid%204%24%2C%20so%20%24p%20%3D%202%24%20is%20regular.%20If%20%24p%20%3D%203%24%20then%20%24%5Coverline%7Bf%7D%20%3D%20x%5E3%20%2B%201%20%3D%20(x%2B1)%5E3%24%2C%20so%20it%20equals%20%24%5Coverline%7Bg_1%7D%5E3%24%20where%20%24g_1%20%3D%20x%2B1%24%2C%20remainder%20%24r_1%20%3D%20f(-1)%20%3D%20-3%20%5Cnmid%209%24.%20Notice%20%24f'%20%3D%203x%5E2%24%2C%20so%20the%20only%20possible%20way%20for%20%24f%24%20to%20have%20double%20root%20is%20when%20the%20root%20is%20%240%24%20(%24p%20%3D%202%24)%20or%20%24f'%20%3D%200%24%20(%24p%3D%203%24).%20Since%20these%20two%20cases%20are%20checked%2C%20%24%5Cmathbb%7BZ%7D%5B%5Csqrt%5B3%5D%7B2%7D%5D%24%20is%20Dedekind%3B%0A%20%20%20%202.%20Consider%20%24f%20%3D%20x%5E5-7%24.%20If%20%24p%20%3D%205%24%20then%20%24%5Coverline%7Bf%7D%20%3D%20(x%2B3)%5E5%24%2C%20%24r_1%20%3D%20(-3)%5E5-7%20%3D%20250%20%5Cequiv%200%20%5Cpmod%7B25%7D%24%2C%20so%20%24P_1%20%3D%20(5%2C%20%5Calpha%2B3)%24%20is%20not%20invertible%2C%20implies%20%24%5Cmathbb%7BZ%7D%5B%5Csqrt%5B5%5D%7B7%7D%5D%24%20is%20not%20Dedekind%3B%0A%20%20%20%203.%20Consider%20%24f%20%3D%20x%5E3-x-1%24%2C%20then%20%24f'%20%3D%203x%5E2-1%24%2C%20%24f%24%20has%20double%20root%20at%20%24x%24%20only%20if%20both%20%24f%24%20and%20%24f'%24%20has%20%24x%24%20as%20a%20root%2C%20so%20that%20%24f%20-%20%5Cfrac%7B1%7D%7B3%7Dxf'%24%20%24%3D%20x%5E3-x-1%20-%20%5Cfrac%7B1%7D%7B3%7Dx(3x%5E2-1)%24%20%24%3D%20-%5Cfrac%7B2%7D%7B3%7Dx-1%20%3D0%24%20%24%5Cimplies%20x%20%3D%20-%5Cfrac%7B3%7D%7B2%7D%24.%20Since%20%24f'(-%5Cfrac%7B3%7D%7B2%7D)%20%3D%20%5Cfrac%7B23%7D%7B4%7D%24%2C%20%24f%24%20attains%20double%20roots%20only%20possibly%20when%20%24p%20%3D%2023%24%20(because%20only%20in%20this%20case%20%24f'(-%5Cfrac%7B3%7D%7B2%7D)%24%20is%20actually%20%240%24).%20If%20%24p%20%3D%2023%24%2C%20then%20%24%5Coverline%7Bf%7D%20%3D%20(x%2B20)(x%2B13)%5E2%24%2C%20consider%20%24g%20%3D%20(x%2B13)%24%2C%20then%20the%20corresponding%20remainder%20is%20%24f(-13)%20%3D%20-2185%20%5Cnot%5Cequiv%200%20%5Cpmod%7B23%5E2%7D%24%2C%20thus%20%24p%20%3D%2023%24%20is%20regular%2C%20%24%5Cmathbb%7BZ%7D%5B%5Calpha%5D%24%20is%20Dedekind.%0A%0A%20%20%20%20%23%23%20Exercise%209%0A%0A%20%20%20%20**Show%20that%20the%20number%20ring%20%24R%24%20is%20a%20UFD%20if%20and%20only%20if%20it%20is%20a%20PID**.%0A%0A%20%20%20%20Proof%0A%0A%20%20%20%20Every%20PID%20is%20UFD%2C%20so%20we%20only%20need%20the%20other%20direction.%0A%0A%20%20%20%20Suppose%20now%20the%20number%20ring%20%24R%24%20is%20a%20UFD%2C%20suppose%20%24P%24%20is%20a%20prime%20ideal%20of%20%24R%24.%20Suppose%20%24x%20%5Cin%20P%24.%20Write%20%24x%24%20as%20product%20of%20irreducibles%2C%20then%20(at%20least)%20one%20of%20the%20irreducible%20factors%20is%20in%20%24P%24%2C%20say%20it%20is%20%24p%24.%20Since%20%24R%24%20is%20a%20UFD%2C%20%24(p)%24%20must%20be%20a%20prime%20ideal%20(any%20element%20in%20%24(p)%24%20is%20of%20the%20form%20%24rp%24%2C%20since%20%24p%24%20is%20irreducible%2C%20at%20least%20one%20of%20terms%20in%20the%20factorization%20of%20%24rp%24%20is%20in%20%24(p)%24).%20We%20know%20in%20a%20number%20ring%20prime%20ideals%20are%20maximal%2C%20so%20%24(p)%24%20is%20maximal.%20But%20of%20course%20%24(p)%20%5Csubseteq%20P%24%2C%20since%20%24P%24%20is%20by%20definition%20proper%2C%20we%20must%20have%20%24(p)%20%3D%20P%24%2C%20in%20particular%20%24P%24%20is%20principal.%20Thus%20any%20prime%20ideal%20is%20principal.%20We%20also%20know%20number%20ring%20is%20noetherian%2C%20so%20by%20Exercise%202%2C%20%24R%24%20is%20PID.%0A%20%20%20%20%22%22%22)%0A%20%20%20%20return%0A%0A%0A%40app.cell%0Adef%20_(mo)%3A%0A%20%20%20%20mo.md(r%22%22%22%0A%20%20%20%20%23%20Spring%20Semester%20Homework%205%0A%20%20%20%20%22%22%22)%0A%20%20%20%20return%0A%0A%0A%40app.cell%0Adef%20_(mo)%3A%0A%20%20%20%20mo.md(r%22%22%22%0A%20%20%20%20%23%23%20Exercise%201%0A%0A%20%20%20%20**Let%20%24d%24%20be%20an%20integer%20that%20is%20not%20a%20cube**.%0A%0A%20%20%20%201.%20**Show%20that%20%24R%20%3D%20%5Cmathbb%7BZ%7D%5B%5Csqrt%5B3%5D%7Bd%7D%5D%24%20is%20a%20Dedekind%20domain%20if%20and%20only%20if%20%24d%24%20is%20square-free%20and%20%24d%20%5Cne%20%5Cpm%201%20%5Cpmod%7B9%7D%24**%3B%0A%20%20%20%202.%20**Let%20now%20%24d%20%3D%20p%24%20be%20prime%2C%20and%20%24d%20%5Cequiv%20%5Cpm%201%20%5Cpmod%7B9%7D%24.%20Determine%20the%20singular%20prime%20ideals%20of%20%24R%24**.%0A%0A%20%20%20%20Proof%0A%0A%20%20%20%201.%20Minimal%20polynomial%20of%20%24%5Csqrt%5B3%5D%7Bd%7D%24%20is%20%24f(x)%20%3D%20x%5E3-d%20%3D%200%24%2C%20so%20%24f'(x)%20%3D%203x%5E2%24.%20The%20possible%20%24p%24%20that%20are%20singular%3A%0A%20%20%20%20%20%20%20%201.%20%24p%20%3D%203%24%2C%20so%20that%20%24%5Coverline%7Bf'%7D%20%3D%200%24.%20Then%20%24%5Coverline%7Bf%7D%20%3D%20(x-d)%5E3%24.%20It%20is%20singular%20if%20the%20remainder%20%24%3Df(d)%24%20%24%3D%20d%5E3-d%20%5Cequiv%200%20%5Cpmod%7B9%7D%24%2C%20we%20can%20check%20that%20this%20happens%20if%20and%20only%20if%20%24d%20%3D%20%5Cpm%201%20%5Cpmod%7B9%7D%24%3B%0A%20%20%20%20%20%20%20%202.%20%24p%24%20such%20that%20%240%24%20is%20a%20double%20root.%20It%20is%20singular%20if%20the%20remainder%20%24%3Df(0)%20%3D%20-d%20%5Cequiv%200%5Cpmod%7Bp%5E2%7D%24%2C%20in%20particular%20if%20%24d%24%20is%20not%20square-free.%0A%20%20%20%20%20%20%20%20Since%20%24R%24%20is%20Dedekind%20if%20and%20only%20if%20there%20is%20no%20singular%20%24p%24%2C%20we%20have%20the%20desired%20result%3B%0A%20%20%20%202.%20By%20Kummer-Dedekind%20Theorem%2C%20all%20prime%20ideals%20of%20%24R%24%20is%20of%20the%20form%20%24(q%2C%20g_i(%5Csqrt%5B3%5D%7Bd%7D))%24%20(using%20the%20notation%20from%20class%2C%20and%20%24q%24%20is%20prime%20integer).%20So%20the%20main%20task%20is%20to%20find%20all%20singular%20prime%2C%20the%20rest%20will%20be%20straight-forward.%0A%20%20%20%20%20%20%20%201.%20By%20above%20we%20know%20that%20%243%24%20must%20be%20a%20singular%20prime%20(given%20%24d%20%5Cequiv%201%20%5Cpmod%7B9%7D%24)%2C%20this%20gives%20a%20singular%20prime%20ideal%20%24(3%2C%20%5Csqrt%5B3%5D%7Bd%7D-d)%24%3B%0A%20%20%20%20%20%20%20%202.%20From%20above%20second%20case%2C%20%24q%24%20is%20singular%20(prime)%20only%20if%20%24d%20%3D%20p%20%5Cequiv%200%20%5Cpmod%7Bq%5E2%7D%24%2C%20but%20that%20is%20not%20possible%20as%20%24p%24%20is%20prime.%0A%20%20%20%20%20%20%20%20So%20there%20is%20only%20one%20singular%20prime%20ideal%20%24(3%2C%20%5Csqrt%5B3%5D%7Bd%7D-d)%24%20in%20%24R%24.%0A%0A%20%20%20%20%23%23%20Exercise%202%0A%20%20%20%20**Let%20%24K%20%3D%20%5Cmathbb%7BQ%7D(%5Calpha)%24%2C%20where%20%24%5Calpha%5E3-3%5Calpha%2B9%20%3D%200%24%2C%20and%20%24R%20%3D%20%5Cmathbb%7BZ%7D%5B%5Calpha%5D%24**.%0A%0A%20%20%20%201.%20**What%20are%20the%20singular%20primes%20in%20%24R%24%3F%20Are%20there%20any%20ramified%20regular%20primes**%3F%0A%20%20%20%202.%20**Describe%20the%20ring%20of%20integers%20%24O_K%24.%20What%20is%20%24%5BO_K%3AR%5D%24**%3F%0A%0A%20%20%20%20Solution%0A%0A%20%20%20%201.%20Denote%20%24f(x)%20%3D%20x%5E3-3x%2B9%24%2C%20then%20%24f'%20%3D%203x%5E2-3%24.%20The%20possible%20%24p%24%20that%20are%20singular%3A%0A%20%20%20%20%20%20%20%201.%20%24p%20%3D%203%24%2C%20in%20this%20case%20%24%5Coverline%7Bf%7D%20%3D%20x%5E3%24%2C%20remainder%20is%20%24f(0)%20%3D%209%20%5Cequiv%200%20%5Cpmod%7B9%7D%24%20so%20this%20is%20indeed%20singular%3B%0A%20%20%20%20%20%20%20%202.%20Otherwise%20consider%20%24f%20-%20%5Cfrac%7B1%7D%7B3%7Dxf'%20%3D%20x%5E3-3x%2B9-x%5E3%2Bx%20%3D%20-2x%2B9%24.%20If%20%24x%24%20is%20both%20root%20for%20%24f%24%20and%20%24f'%24%20(so%20that%20it%20is%20double%20root%20for%20%24f%24)%20then%20it%20must%20be%20root%20of%20%24f-%5Cfrac%7B1%7D%7B3%7Dxf'%24%2C%20in%20particular%20%24x%20%3D%20%5Cfrac%7B9%7D%7B2%7D%24%20is%20the%20only%20possible%20double%20root.%20Since%20%24f'(%5Cfrac%7B9%7D%7B2%7D)%20%3D%20231%2F4%24%20where%20%24231%20%3D%203%20%5Ccdot%207%20%5Ccdot%2011%24%2C%20the%20possible%20%24p%24%20that%20really%20make%20%24%5Cfrac%7B9%7D%7B2%7D%24%20being%20a%20root%20of%20%24f'%24%20are%20%243%24%2C%20%247%24%2C%20or%20%2411%24.%20We%20already%20see%20%243%24%20is%20singular%2C%20so%20we%20check%20the%20other%20two%3A%0A%20%20%20%20%20%20%20%20%20%20%20%201.%20If%20%24p%20%3D%207%24%20then%20%24%5Coverline%7Bf%7D%20%3D%20x%5E3-3x%2B2%24%20%24%3D%24%20%24(x%2B2)(x%2B6)%5E2%24%2C%20%24-6%24%20is%20a%20double%20root%2C%20and%20the%20remainder%20%24f(-6)%20%3D%20-189%20%5Cequiv%207%20%5Cpmod%7B49%7D%24%2C%20so%20%24p%20%3D%207%24%20is%20not%20singular%2C%20it%20is%20ramified%20regular%3B%0A%20%20%20%20%20%20%20%20%20%20%20%202.%20If%20%24p%20%3D%2011%24%20then%20%24%5Coverline%7Bf%7D%20%3D%20(x%2B9)(x%2B1)%5E2%24%2C%20%24-1%24%20is%20a%20double%20root%2C%20and%20remainder%20%24f(-1)%20%3D%2011%20%5Cequiv%2011%20%5Cpmod%7B121%7D%24%2C%20so%20%2411%24%20is%20also%20a%20ramified%20regular.%0A%20%20%20%202.%20We%20saw%20above%20that%20the%20only%20singular%20prime%20of%20%24R%24%20is%20%243%24%2C%20the%20corresponding%20prime%20ideal%20is%20%24(3%2C%20%5Calpha)%24.%20Consider%20%24%5Calpha%5E3-3%5Calpha%2B9%20%3D%20%5Calpha%5Ccdot(%5Calpha%5E2-3)%2B9%20%5Cimplies%20-%5Cfrac%7B%5Calpha%5E2-3%7D%7B3%7D%20%3D%20%5Cfrac%7B3%7D%7B%5Calpha%7D%24%2C%20say%20they%20equal%20%24%5Cbeta%24%2C%20then%20%24%5Cbeta%20%5Cnotin%20R%24%20(otherwise%20%24(3%2C%5Calpha)%24%20is%20principal)%2C%20then%20%24%5Cfrac%7B3%7D%7B%5Cbeta%7D%20%3D%20%5Calpha%20%5Cimplies%20(%5Cfrac%7B3%7D%7B%5Cbeta%7D)%5E3%20-%203(%5Cfrac%7B3%7D%7B%5Cbeta%7D)%2B9%20%3D%200%24%20%24%5Cimplies%20%5Cbeta%5E3-%5Cbeta%5E2%2B3%20%3D%200%24.%20This%20is%20an%20irreducible%20polynomial%20(thus%20is%20minimal%20polynomial%20of%20%24%5Cbeta%24).%20We%20now%20want%20to%20find%20singular%20primes%20in%20%24%5Cmathbb%7BZ%7D%5B%5Calpha%2C%5Cbeta%5D%24.%20If%20%24g%20%3D%20x%5E3-x%5E2%20%2B%203%24%20then%20%24g'%20%3D%203x%5E2-2x%24.%0A%20%20%20%20%20%20%20%20Proceed%20as%20before%2C%20if%20%24g(x)%20%3D%200%24%20and%20%24g'(x)%20%3D%200%24%20then%20%24(g-%20%5Cfrac%7Bx%7D%7B3%7Dg')(x)%20%3D%200%24%2C%20which%20means%24(x-3)(x%2B3)%20%3D%200%24%20thus%20%24x%20%3D%20%5Cpm%203%24.%20In%20particular%20%24p%20%3D%203%24%20is%20the%20only%20possible%20singular%20prime%2C%20but%20since%20%24g(0)%20%3D%203%20%5Cequiv%203%20%5Cpmod%7B9%7D%24%2C%20it%20is%20ramified%20regular.%20So%20%24%5Cmathbb%7BZ%7D%5B%5Calpha%2C%5Cbeta%5D%24%20is%20Dedekind%20so%20%24%5Cmathbb%7BZ%7D%5B%5Calpha%2C%5Cbeta%5D%20%5Csupseteq%20O_K%24.%20By%20construction%20%24%5Cbeta%24%20is%20an%20algebraic%20integer%2C%20we%20also%20have%20%24O_K%20%5Csupseteq%20%5Cmathbb%7BZ%7D%5B%5Calpha%2C%5Cbeta%5D%24%2C%20and%20thus%20%24O_K%20%3D%20%5Cmathbb%7BZ%7D%5B%5Calpha%2C%5Cbeta%5D%24.%0A%20%20%20%20%20%20%20%20Basis%20of%20%24R%24%20is%20%24%5Clbrace%201%2C%20%5Calpha%2C%20%5Calpha%5E2%20%5Crbrace%24%2C%20and%20basis%20of%20%24R'%20%3D%20%5Cmathbb%7BZ%7D%5B%5Calpha%2C%5Cbeta%5D%24%20%24%3D%20%5Cmathbb%7BZ%7D%5B%5Calpha%2C%20-%5Cfrac%7B%5Calpha%5E2-3%7D%7B3%7D%5D%24%20is%20%24%5Clbrace%201%2C%20%5Calpha%2C%20-%5Cfrac%7B%5Calpha%5E2-3%7D%7B3%7D%5Crbrace%24%2C%20so%20a%20linear%20transformation%20%24T%3A%20R%20%5Cto%20R'%24%20can%20be%20described%20as%20%24%5Cbegin%7Bpmatrix%7D1%260%261%5C%5C0%261%260%5C%5C0%260%26%5Cfrac%7B1%7D%7B3%7D%5Cend%7Bpmatrix%7D%24%2C%20and%20%24%5BR'%3AR%5D%20%3D%20%7C%20%5Cdet%7BT%5E%7B-1%7D%7D%20%7C%20%3D%203%24.%0A%0A%20%20%20%20%23%23%20Exercise%203%0A%0A%20%20%20%20**Suppose%20%24L%20%3D%20%5Cmathbb%7BQ%7D(%5Calpha)%24%2C%20and%20%24f%24%20is%20the%20minimal%20polynomial%20of%20%24%5Calpha%24%20over%20%24%5Cmathbb%7BQ%7D%24**.%0A%0A%20%20%20%201.%20**Verify%20that%20%24N_%7BL%2F%5Cmathbb%7BQ%7D%7D(c-%5Calpha)%20%3D%20f(c)%24%20for%20every%20%24c%20%5Cin%20%5Cmathbb%7BQ%7D%24**%3B%0A%20%20%20%202.%20**Assume%20now%20that%20%24p%20%5Cequiv%201%20%5Cpmod%7B4%7D%24%2C%20and%20prove%20that%20%24%5Cfrac%7Bp%5Ep-1%7D%7Bp-1%7D%24%20is%20not%20prime**.%20*Hint%3A%20In%20one%20of%20the%20homework%20last%20semester%2C%20we%20checked%20that%20for%20odd%20%24p%24%20the%20cyclotomic%20field%20contains%20an%20element%20%24%5Ctau%24%20such%20that%20%24%5Ctau%5E2%20%3D%20%5Cpm%20p%24*.%0A%0A%20%20%20%20Proof%0A%0A%20%20%20%201.%20We%20know%20%24f(c)%20%3D%20f_%7BL%2F%5Cmathbb%7BQ%7D%7D%5E%7B%5Calpha%7D(c)%24%20for%20every%20%24c%20%5Cin%20%5Cmathbb%7BQ%7D%24%2C%20which%20by%20definition%20equals%20%24%5Cdet(c%5Ccdot%20%5Ctext%7Bid%7D_L%20-%20M_%7B%5Calpha%7D)%24%20which%20is%20exactly%20%24%5Cdet(M_%7Bc-%5Calpha%7D)%24%20(because%20for%20any%20%24y%20%5Cin%20L%24%2C%20%24(c%5Ccdot%20%5Ctext%7Bid%7D_L%20-%20M_%7B%5Calpha%7D)(y)%20%3D%20cy-%5Calpha%20y%20%3D%20(c-%5Calpha)(y)%20%3D%20M_%7Bc-%5Calpha%7D(y)%24)%20which%20is%20by%20definition%20%24N_%7BL%2F%5Cmathbb%7BQ%7D%7D(c-%5Calpha)%24%3B%0A%20%20%20%202.%20Consider%20%24%5Calpha%20%3D%20e%5E%7B2%5Cpi%20i%2Fp%7D%24%20so%20that%20%24L%20%3D%20Q_p%24%20is%20the%20%24p%24-th%20cyclotomic%20field.%0A%20%20%20%20%20%20%20%201.%20Suppose%20%24p%5Cequiv%201%20%5Cpmod%7B4%7D%24%20but%20%24p%24%20is%20not%20prime.%20Then%20we%20know%20from%20last%20semester%20that%20%24x%5Ep-1%20%3D%20%5Cprod%5Climits_%7Bd%7C%20p%7D%5CPhi_d(x)%24%20%24%3D(x-1)%5Cprod%5Climits_%7Bd%7C%20p%2C%20d%20%5Cne%201%7D%5CPhi_d(x)%24%20thus%20%24%5Cfrac%7Bx%5Ep-1%7D%7Bx-1%7D%20%3D%5Cprod_%7Bd%20%7C%20p%2C%20d%5Cne%201%7D%5CPhi_d(x)%24%20where%20%24%5CPhi_d(x)%20%5Cin%20%5Cmathbb%7BZ%7D%5Bx%5D%24.%20Since%20%24p%24%20is%20assumed%20not%20to%20be%20prime%2C%20it%20has%20at%20least%20%242%24%20factors%20that%20is%20not%20%241%24%2C%20so%20%24%5Cfrac%7Bx%5Ep-1%7D%7Bx-1%7D%24%20is%20product%20of%20at%20least%20two%20%24%5CPhi_d(x)%24.%20Plugin%20%24x%20%3D%20p%24%20we%20get%20that%20%24%5Cfrac%7Bp%5Ep-1%7D%7Bp-1%7D%24%20has%20at%20least%20two%20factors%20(they%20are%20clearly%20distinct%20and%20not%20%241%24)%20so%20it%20is%20not%20prime%3B%0A%20%20%20%20%20%20%20%202.%20Otherwise%20%24p%20%5Cequiv%201%20%5Cpmod%7B4%7D%24%20and%20%24p%24%20is%20prime.%20From%20homework%20in%20last%20semester%20we%20know%20that%20there%20is%20some%20%24%5Ctau%20%5Cin%20%5Cmathbb%7BQ%7D_p%24%20such%20that%20%24%5Ctau%5E2%20%3D%20p%24.%20Since%20%24p%24%20is%20prime%2C%20%24f%20%3D%20%5Cfrac%7Bx%5Ep-1%7D%7Bx-1%7D%24%20is%20minimal%20polynomial%20for%20both%20%24%5Calpha%24%20and%20%24%5Calpha%5E2%24.%20So%20by%20part%201%2C%20we%20get%20%24N_%7BL%2F%5Cmathbb%7BQ%7D%7D(p-%5Calpha)%20%3D%20f(p)%24%20and%20%24N_%7BL%2F%5Cmathbb%7BQ%7D%7D(p-%5Calpha%5E2)%20%3D%20f(p)%24%2C%20so%20%24%24N_%7BL%2F%5Cmathbb%7BQ%7D%7D(p-%5Calpha)%20%3D%20N_%7BL%2F%5Cmathbb%7BQ%7D%7D((%5Ctau%2B%5Calpha)(%5Ctau-%5Calpha))%20%3D%20N_%7BL%2F%5Cmathbb%7BQ%7D%7D(%5Ctau%2B%5Calpha)N_%7BL%2F%5Cmathbb%7BQ%7D%7D(%5Ctau-%5Calpha).%24%24%20Since%20%24%5Ctau%24%20(and%20%24%5Calpha%24%2C%20of%20course)%20is%20in%20%24O_L%24%2C%20we%20have%20both%20%24N_%7BL%2F%5Cmathbb%7BQ%7D%7D(%5Ctau%2B%5Calpha)%2CN_%7BL%2F%5Cmathbb%7BQ%7D%7D(%5Ctau-%5Calpha)%24%20are%20integers.%20We%20have%20the%20desired%20statement%20if%20they%20are%20both%20not%20%241%24%3A%0A%20%20%20%20%20%20%20%203.%20If%20%24N_%7BL%2F%5Cmathbb%7BQ%7D%7D(%5Ctau-%5Calpha)%20%3D%201%24%2C%20then%20%24%5Cfrac%7B%5Ctau%5Ep-1%7D%7B%5Ctau-1%7D%20%3D%201%20%5Cimplies%20%5Ctau%5Ep%20%3D%20%5Ctau%24%20%24%5Cimplies%20%5Ctau%5E%7Bp-1%7D%20%3D%201%24%20which%20cannot%20be%20true%3B%0A%20%20%20%20%20%20%20%204.%20If%20%24N_%7BL%2F%5Cmathbb%7BQ%7D%7D(%5Ctau%2B%5Calpha)%20%3D%201%24%20then%20%24N_%7BL%2F%5Cmathbb%7BQ%7D%7D(%5Ctau-%5Calpha)%20%3D%20(%5Cfrac%7B%5Ctau%5Ep-1%7D%7B%5Ctau-1%7D)%2F1%20%3D%20%5Cfrac%7B%5Ctau%5Ep-1%7D%7B%5Ctau-1%7D%24%20%24%5Cimplies%24%20%24%5Ctau%5E%7Bp-1%7D%20%3D%201%24%20again.%0A%0A%20%20%20%20%20%20%20%20So%20they%20are%20both%20not%20%241%24%20and%20we%20get%20the%20desired%20statement.%0A%0A%20%20%20%20%23%23%20Exercise%204%0A%0A%20%20%20%20**Compute%20the%20class%20group%20%24%5Ctext%7BCl%7D(L)%24%2C%20where**%3A%0A%0A%20%20%20%201.%20**%24L%20%3D%20%5Cmathbb%7BQ%7D(%5Csqrt%7B-71%7D)%24**%3B%0A%20%20%20%202.%20**%24L%20%3D%20%5Cmathbb%7BQ%7D(%5Csqrt%7B-65%7D)%24**.%0A%0A%20%20%20%20Solution%0A%0A%20%20%20%201.%20%24-71%20%5Cequiv%201%20%5Cpmod%7B4%7D%24%2C%20so%20%24O_L%20%3D%20%5Cmathbb%7BZ%7D%5B%5Cfrac%7B1%2B%5Csqrt%7B-71%7D%7D%7B2%7D%20%3D%20%5Calpha%5D%24.%20Minimal%20polynomial%20of%20%24%5Calpha%24%20is%20%24f(x)%20%3D%20x%5E2-x%2B18%24.%20Take%20%24c%20%5Cin%20%5Cmathbb%7BZ%7D%24%2C%20%24N(c-%5Calpha)%24%20%24%3D%20f(c)%24.%20In%20particular%20we%20can%20calculate%3A%20%24%24%5Cbegin%7Barray%7D%7B%7Cr%7Cr%7C%7D%5Chline%20c%26f(c)%20%3D%20N(c-%5Calpha)%20%5C%5C%20%5Chline%20-1%262%5E2%5Ccdot5%5C%5C%5Chline%200%262%5Ccdot%203%5E2%5C%5C%5Chline%201%262%5Ccdot%203%5E2%5C%5C%5Chline%202%262%5E2%5Ccdot%205%5C%5C%5Chline%203%262%5E3%5Ccdot%203%5C%5C%5Chline%5Cend%7Barray%7D%24%24%0A%0A%20%20%20%20%20%20%20%201.%20If%20%24p%20%3D%202%24%2C%20then%20%24%5Coverline%7Bf%7D%20%3D%20x(x-1)%24%2C%20%24(2)%20%3D%20P_2Q_2%24%20where%20%24P_2%20%3D%20(2%2C%5Calpha)%24%2C%20%24Q_2%20%3D%20(2%2C%20%5Calpha-1)%24%3B%0A%20%20%20%20%20%20%20%202.%20If%20%24p%20%3D%203%24%2C%20then%20%24%5Coverline%7Bf%7D%20%3D%20x(x-1)%24%2C%20%24(3)%24%20%24%3D%20P_3Q_3%24%20where%20%24P_3%20%3D%20(3%2C%5Calpha)%24%2C%20%24Q_3%20%3D%20(3%2C%20%5Calpha-1)%24%3B%0A%20%20%20%20%20%20%20%203.%20If%20%24p%20%3D%205%24%2C%20then%20%24%5Coverline%7Bf%7D%20%3D%20(x%2B1)(x%2B3)%24%2C%20%24(7)%20%3D%20P_7Q_7%24%20where%20%24P_5%20%3D%20(5%2C%5Calpha%2B1)%24%2C%20%24Q_5%20%3D%20(5%2C%5Calpha%2B3)%24.%0A%0A%20%20%20%20%20%20%20%20In%20the%20class%20group%2C%20%24%5BP_2%5D%20%3D%20-%5BQ_2%5D%24%2C%20%24%5BP_3%5D%20%3D%20-%5BQ_3%5D%24%20and%20%24%5BP_5%5D%20%3D%20-%5BQ_5%5D%24.%20Read%20the%20table%20we%20also%20have%3A%0A%0A%20%20%20%20%20%20%20%201.%20%24(-1-%5Calpha)%20%3D%20Q_2%5E2P_5%24%20thus%20%242%5BQ_2%5D%20%3D%20-%5BP_5%5D%24%3B%0A%20%20%20%20%20%20%20%202.%20%24(%5Calpha)%20%3D%20P_2P_3%5E2%24%20thus%20%24%5BP_2%5D%20%3D%20-2%5BP_3%5D%24%3B%0A%20%20%20%20%20%20%20%203.%20%24(3-%5Calpha)%20%3D%20Q_2%5E3P_3%24.%0A%0A%20%20%20%20%20%20%20%20If%20we%20set%20%24%5BP_3%5D%20%3D%20%5Ceta%20%5Cin%20%5Ctext%7BCl%7D(L)%24%2C%20then%20we%20get%20%24%5BP_2%5D%20%3D%20-2%5Ceta%24%20and%20%24%5BP_5%5D%3D%20-4%5Ceta%24%2C%20also%20from%203.%20We%20get%20%247%5Ceta%20%3D%200%24.%20Thus%20a%20reasonable%20guess%20would%20be%20%24%5Ctext%7BCl%7D(L)%20%5Ccong%20%5Cmathbb%7BZ%7D_7%20%3D%20%5Clangle%20%5Ceta%20%5Crangle%24.%20We%20need%20to%20check%20if%20%24P_3%24%20is%20principal.%20If%20%24P_3%20%3D%20(3%2C%5Calpha)%20%3D%20(%5Cdelta)%24%2C%20then%20%24N_%7BL%2F%5Cmathbb%7BQ%7D%7D(%5Cdelta)%20%3D%20%5Cdelta%5Coverline%7B%5Cdelta%7D%20%3D%203%24%2C%20but%20for%20%24%5Cdelta%20%5Cin%20%5Cmathbb%7BZ%7D%5B%5Cfrac%7B1%2B%5Csqrt%7B-71%7D%7D%7B2%7D%5D%24%20this%20is%20not%20possible.%20So%20%24%5Ctext%7BCl%7D(L)%20%5Ccong%20%5Cmathbb%7BZ%7D_7%24.%0A%20%20%20%202.%20%24-65%20%5Cequiv%203%20%5Cpmod%7B4%7D%24%2C%20so%20%24O_L%20%3D%20%5Cmathbb%7BZ%7D%5B%5Csqrt%7B-65%7D%5D%24%2C%20corresponding%20minimal%20polynomial%20is%20%24f%20%3D%20x%5E2%20%2B%2065%24.%20We%20can%20calculate%20%24M_R%20%3D%20%5Cfrac%7B2%7D%7B%5Cpi%7D%5Csqrt%7B260%7D%24%20%24%5Capprox%2010%24%20so%20we%20need%20to%20look%20over%20%24p%20%3D%202%2C3%2C5%2C7%24%3A%0A%20%20%20%20%20%20%20%201.%20%24p%20%3D%202%24%2C%20then%20%24%5Coverline%7Bf%7D%20%3D%20(x%2B1)%5E2%24%2C%20%24(2)%20%3D%20P_2%5E2%24%20where%20%24P_2%20%3D%20(2%2C%20%5Calpha%2B1)%24%2C%20%242%5BP_2%5D%20%3D%200%24%3B%0A%20%20%20%20%20%20%20%202.%20%24p%20%3D%203%24%2C%20then%20%24%5Coverline%7Bf%7D%20%3D%20(x%2B1)(x%2B2)%24%2C%20%24(3)%20%3D%20P_3Q_3%24%20where%20%24P_3%20%3D%20(3%2C%5Calpha%2B1)%24%2C%20%24Q_3%3D%20(3%2C%20%5Calpha%2B2)%24%2C%20%24%5BP_3%5D%20%3D%20-%5BQ_3%5D%24%3B%0A%20%20%20%20%20%20%20%203.%20%24p%20%3D%205%24%2C%20then%20%24%5Coverline%7Bf%7D%20%3D%20x%5E2%24%2C%20%24(5)%20%3D%20P_5%5E2%24%20where%20%24P_5%20%3D%20(5%2C%5Calpha)%24%2C%20%242%5BP_5%5D%20%3D%200%24%3B%0A%20%20%20%20%20%20%20%204.%20%24p%20%3D%207%24%2C%20then%20%24%5Coverline%7Bf%7D%20%3D%20x%5E2%2B2%24%20is%20inert.%0A%0A%20%20%20%20%20%20%20%20Consider%20%24%5Cbeta%20%3D%205-%20%5Calpha%24%2C%20then%20%24N(%5Cbeta)%20%3D%20f(5)%20%3D%2090%20%3D%202%5Ccdot%203%5E2%20%5Ccdot%205%24%2C%20so%20%24(5-%5Calpha)%20%3D%20P_2P_3%5E2P_5%24%20in%20particular%20%24%5BP_2P_5%5D%20%3D%20-2%5BP_3%5D%24.%20%24P_2P_5%24%20cannot%20be%20principal%2C%20because%20we%20can%20calculate%20that%20there%20is%20no%20ideal%20of%20norm%20%2410%24.%0A%20%20%20%20%20%20%20%20Consider%20%24%5Cgamma%20%3D%204-%5Calpha%24%2C%20then%20%24N(%5Cgamma)%20%3D%20f(4)%24%20%24%3D%2081%20%3D%203%5E4%24%2C%20so%20%244%5BP_3%5D%20%3D%200%24%2C%20%24%5BP_3%5D%24%20is%20of%20order%20%244%24.%0A%20%20%20%20%20%20%20%20Also%20%24%5BP_2%5D%20%5Cne%20-%5BP_3%5D%24%2C%20otherwise%20%24P_2P_3%24%20is%20principal%20but%20there%20is%20no%20ideal%20of%20norm%20%246%24.%20So%20we%20can%20use%20%24%5BP_2%5D%24%20and%20%24%5BP_3%5D%24%20as%20generator%2C%20we%20get%20%24%5Ctext%7BCl%7D(L)%20%3D%20%5Cmathbb%7BZ%7D_2%20%5Ctimes%20%5Cmathbb%7BZ%7D_4%24.%0A%0A%20%20%20%20%23%23%20Exercise%205%0A%0A%20%20%20%20**Let%20%240%20%5Cto%20K%20%5Cto%20P%20%5Cxrightarrow%7B%5Cvarphi%7D%20M%20%5Cto%200%24%20and%20%240%20%5Cto%20K'%20%5Cto%20P'%20%5Cxrightarrow%7B%5Cvarphi'%7D%20M%20%5Cto%200%24%20be%20two%20exact%20sequences%20of%20%24R%24-modules%20with%20%24P%24%2C%20%24P'%24%20both%20projective.%20Prove%20that%20%24P%20%5Coplus%20K'%20%5Ccong%20P'%20%5Coplus%20K%24%20as%20%24R%24-modules**.%20*Hint%3A%20consider%20the%20module%20given%20by%20the%20condition%20%24%5Cvarphi(p)%20%3D%20%5Cvarphi'(p')%24*.%0A%0A%20%20%20%20Proof%0A%0A%20%20%20%20Consider%20the%20sub-module%20%24N%20%5Csubset%20P%20%5Coplus%20P'%24%20by%20%24N%20%3D%20%5Clbrace%20(p%2C%20p')%20%7C%20%5Cvarphi(p)%20%3D%20%5Cvarphi'(p')%20%5Crbrace%24.%20Claim%2C%20for%20each%20%24p%20%5Cin%20P%24%2C%20there%20exists%20%24p'%20%5Cin%20P'%24%20so%20that%20%24(p%2Cp')%24%20is%20in%20%24N%24%2C%20and%20vice%20versa.%20This%20is%20easy%3A%20for%20any%20given%20%24p%24%2C%20let%20%24%5Cvarphi(p)%20%3D%20m%24%2C%20since%20%24%5Cvarphi'%24%20in%20surjective%2C%20we%20can%20find%20%24p'%24%20so%20that%20%24%5Cvarphi'(p')%20%3D%20m%24%20so%20%24%5Cvarphi'(p')%20%3D%20%5Cvarphi(p)%24.%0A%0A%20%20%20%20Now%20consider%20map%20%24%5Calpha%3A%20N%20%5Cto%20P%24%20given%20by%20%24(p%2Cp')%20%5Cto%20p%24.%20By%20above%20discussion%2C%20this%20map%20is%20surjective.%20Thus%20by%20a%20proposition%20from%20class%2C%20this%20map%20splits%2C%20where%20%24%24%5Cbegin%7Baligned%7D%5Cker%7B%5Calpha%7D%20%26%3D%20%5Clbrace%20(p%2Cp')%20%7C%20p%20%3D%200%2C%5Cvarphi(p)%20%3D%20%5Cvarphi'(p')%5Crbrace%20%5C%5C%20%26%3D%5Clbrace%20(0%2Cp')%20%7C%20%5Cvarphi'(p')%20%3D%200%5Crbrace%20%5C%5C%26%20%5Ccong%20%5Cker%7B%5Cvarphi'%7D%5Cend%7Baligned%7D%24%24%20which%20is%20isomorphic%20to%20%24K'%24%20because%20%240%20%5Cto%20K'%20%5Cto%20P'%20%5Cxrightarrow%7B%5Cvarphi'%7D%20M%20%5Cto%200%24%20is%20assumed%20to%20be%20a%20short%20exact%20sequence.%20In%20particular%2C%20we%20get%20that%20%24%240%20%5Cxrightarrow%7B%7D%20K'%20%5Cxrightarrow%7B%7D%20N%20%5Cxrightarrow%7B%5Calpha%7D%20P%20%5Cxrightarrow%7B%7D%200%24%24%20is%20a%20split%20short%20exact%20sequence%2C%20thus%20%24N%20%5Ccong%20P%20%5Coplus%20K'%24.%0A%0A%20%20%20%20Now%20do%20the%20same%20thing%20with%20the%20map%20%24%5Cbeta%3A%20N%20%5Cto%20P'%24%20given%20by%20%24(p%2Cp')%20%5Cto%20p'%24%2C%20and%20we%20get%20%24N%20%5Ccong%20P'%20%5Coplus%20K%24%2C%20put%20things%20together%20we%20have%20%24P'%20%5Coplus%20K%20%5Ccong%20P%20%5Coplus%20K'%24.%0A%20%20%20%20%22%22%22)%0A%20%20%20%20return%0A%0A%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20app.run()%0A