ALGEBRA PRELIM EXAM August 2005

Let \(G\) be a finite group and \(p\) a prime number such that \(|G|=p^n\) for some \(n>0\). Prove that \(G\) has nontrivial center and that for all integers \(0<m<n\), \(G\) has at least one normal subgroup of order \(p^m\).

Let \(G\) be a group of order 63. Show that \(G\) contains an element of order 21.

Let \(R\) be a commutative ring with 1, \(I,J\) ideals such that \(I+J=R\). Show that \(R/(IJ)\cong R/I\oplus R/J\).

Let \(S\) be a multiplicative subset of commutative ring \(A\) not containing 0. Let \(P\) be a maximal element in the set of ideals of \(A\) whose intersection with \(S\) is empty. Show that \(P\) is prime.

Let \((R,m)\) be a local ring (i.e. a commutative ring with a unique maximal ideal \(m\)). Let \(M\) be a finitely generated \(R\)-module with \(mM=M\). Show from first principles that \(M=\{0\}\).

Find a galois extension \(K\) of \(\mathbb {Q}\) with Galois group \(\mathbb {Z}/5\mathbb {Z}\oplus \mathbb {Z}/5\mathbb {Z}\).

Let \(\alpha \) be a primitive fifth root of 1 and \(f(x)=x^5+5x^2+4\). Show that \(f(x)\) is irreducible in \(\mathbb {Q}[\alpha ]\).

Let \(K\) be a finite field.

1. Show that \(|K|=p^n\) for some prime \(p\).
2. Show that \(K^*=K-\{0\}\) is a cyclic group with respect to multiplication.

Let \(A\) be an \(n\times n\) matrix with complex coefficients. Show that \(A^n=0\) if and only if \(\operatorname {tr}(A^i)=0\) for all \(i>0\).

Suppose \(F\) is a field and \(T\) is an invertible \(n\times n\) matrix with entries in \(F\). Show that there exist constants \(a_1,\ldots ,a_k\) in \(F\) so that \[ T^{-1}=\sum _{i=1}^k a_iT^i. \]