丘成桐数学竞赛2021年决赛笔试真题2021+YCMC+Algebra+problems(S)

Algebra,NumberTheoryandCombinatorics(2021)Problem1.(Individualround.)LetpbeaprimenumberandQpthefi≥1beanintegerandL=Qp(ζpn),whereζinetheimageofthenormmapNL/Qp:L×→Q××mayusetheinequality[L:Qp]≤(Q×p:NL/Qp(L))withoutproofinthecasen≥m2.(Individualround.)Letkbeafierthegrouphomomorphism:φ:GL(V)→GL(∧2V),f→∧2f.(1)Determinethekernelofφ.(2)Showthatφinducesagrouphomomorphismψ:SL(V)→SL(∧2V).Expressdet(∧2f)intermsofdet(f).Problem3.(Individualround.)LetAbearank2integermatrixofsize5×fyallpossiblegroupsoftheformZ5/showthatNL/Qp(L×)=pZ(1+pnZp),−1LetΦ(X)=(Xp−1)/(Xp−1)Φ(X+1)Φ(X)istheminimalpolynomialofζpn,sothatNL/Qp(1−ζpn)=Φ(1)=[L:Qp]=φ(pn)=pn−pn−d,theφ(pn)-thpowermapon1+pZpisthecompositionnn1+pZp−→pZ−−→pZ−→1+pZp.p−p−∼∼∼logφ(pn)expThusNL/Qp(L×)⊇NL/Qp(1+pZp)=1+=2,wemayassumen≥φ(2n)-thpowermapon1+4Z2isthecompositionφ(2n)logexpn+1n+11+4Z2−→4Z−−→2Z−→1+2Z2.2−2−∼∼∼ThusNL/Q2(L×)⊇NL/Q2(1+4Z2)=1+2n+sytoseethat1+2Z2=(1+2and52n−2nn+1Z2)