PRELIMINARY EXAMINATION IN ALGEBRA September 9, 1996

Let \(G\) be a finite group of order \(p^n\) acting on a set \(X\) consisting of \(k\) elements. Show that the number of points fixed by all elements of \(G\) is congruent to \(k\) modulo \(p\).

Find all rings (with 1), up to isomorphism, containing exactly 25 elements.

Show that there are no simple groups of order 520.

Let \(K\subseteq L\) be finite fields. Prove that the extension \(L\) of \(K\) is Galois with cyclic Galois group.

Let \(E\) be a finite extension of the rationals. Show that the number of roots of unity in \(E\) is finite.

Let \(P_\bullet \) be a complex of projective modules with \(P_n=0\) for \(n<0\). Let \(E_\bullet \) be an exact complex. Show that any map of complexes from \(P_\bullet \) to \(E_\bullet \) is homotopic to zero.

Prove the Cayley-Hamilton Theorem (the theorem that states that for every matrix \(A\) we have \(f(A)=0\), where \(f(X)\) is the characteristic polynomial of \(A\)).

Let \(f\colon \mathbb {Z}^n\to \mathbb {Z}^n\) be a map of free Abelian groups of rank \(n\) defined by a matrix \(A\) with non-zero determinant. Show that the cokernel of \(f\) is finite and that its order is equal to the absolute value of the determinant of \(A\).

Let \(K\) be a field of characteristic zero. Let \(L\) be the splitting field of the polynomial \(X^n-a\) for some \(a\in K\) and some integer \(n\ge 1\). Show that the Galois group of \(L\) over \(K\) is solvable.

Let \(I\) and \(J\) be ideals of a commutative ring \(R\). Show that \((R/I)\otimes _R(R/J)\) is isomorphic to \(R/(I+J)\).

Let \(G\) be a finite subgroup of the group \(\operatorname {GL}_n(\mathbb {C})\) of invertible \(n\) by \(n\) matrices with entries in \(\mathbb {C}\).

1. Show that every element of \(G\) is diagonalizable over \(\mathbb {C}\).
2. Give an example of a finite subgroup of \(\operatorname {GL}_n(\mathbb {Q})\) which contains elements which are not diagonalizable over \(\mathbb {Q}\).

Show that if a module \(M\) is a sum of simple modules, then it is a direct sum of simple modules.