PRELIMINARY EXAMINATION IN ALGEBRA August 20, 1998

Let \(q\) be a prime number. Show that the number of non-isomorphic abelian \(q\)-groups of order \(q^n\) is \(p(n)\), where \(p(n)\) denotes the number of partitions of \(n\).

Let \(M\) be a square matrix over \(\mathbb {C}\) such that \(M^k\) is the identity matrix for some integer \(k>0\). Show that \(M\) is diagonalizable.

Let \(k\) be a field containing a primitive \(n^{\mathrm {th}}\) root of unity. Show that for any \(a\in k\) the Galois group of \(X^n-a\) over \(k\) is abelian.

Show that there are no simple groups of order 245.

State and prove the Hilbert Basis Theorem.

Prove Wedderburn’s theorem: if the ring \(R\) has a faithful simple module that is finite-dimensional over its commutant ring, then \(R\cong M_n(D)\) for some \(n\ge 1\), where \(D\) is a division ring.

Prove that a free group has no nonidentity elements of finite order.

Let \(K/k\) be a Galois extension with Galois group \(G\). Show that every 1-cycle on \(G\) with values in \(K^*\) is a 1-coboundary.

If \(H\) is a finite group, prove that there exists a field \(k\) and a Galois extension of \(k\) with Galois group \(H\).

Prove that the Galois group of \(X^3-X-1\) over \(\mathbb {Q}\) is \(S_3\), the symmetric group on three elements.

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