ALGEBRA PRELIM January, 2004

Suppose \(G\) is a finite group of order a power of a prime. If \(H\) is a nonidentity normal subgroup of \(G\), prove that \(H\) contains a nonidentity element of \(Z(G)\). \[ Z(G)=\{x\in G\mid xy=yx\text { for all }y\in G\}. \]

Prove or disprove that \(x^5-x+1\) is irreducible over the rationals.

Let \(V\) be a finite dimensional vector space over the field of rational numbers and let \(T\) be a linear transformation of \(V\) into itself such that \(T^5\) is the identity mapping. If there is no nonzero vector in \(V\) which \(T\) maps to itself, prove that the dimension of \(V\) is a multiple of 4.

Suppose \(H\) is a subgroup of the finite group \(G\) and the index of \(H\) in \(G\) is the smallest prime dividing the order of \(G\). Prove that \(H\) is a normal subgroup of \(G\).

Compute the Galois group of the polynomial \(x^4-2\) over each of these fields: (a) the complex numbers (b) the reals, (c) the rationals, and (d) the integers modulo 5.

Let \(R\) be a ring and let \(M\) be an irreducible \(R\)-module. Prove that one of the following statements is true about \(M\) considered as an abelian group:

1. \(M\) is the direct sum of (perhaps infinitely many) copies of the additive group of all rational numbers.
2. \(M\) is the direct sum of (perhaps infinitely many) copies of a cyclic group of order \(p\) for some prime \(p\).

Let \(R\) be a ring and let \(L\) be the set of all nilpotent elements of \(R\).

1. If \(R\) is commutative, prove that \(L\) is an ideal.
2. Give an example of a ring \(R\) in which \(L\) is not an ideal.

Prove that a group of order 112 cannot be simple.

Let \(R\) be a semisimple right Artinian ring (in particular, this implies that \(R\) has an identity 1). Assume \(x\in R\) and define the sets \(A_x\) and \(B_x\) by \[ A_x=\{y\in R\mid y\ne 0\text { and }xy=0\} \quad \text {and}\quad B_x=\{y\in R\mid xy=1\}. \] Prove that exactly one of these sets is empty.

Let \(E\) be an extension of the field \(F\) and let \(\alpha \) be an element of \(E\) which is algebraic over \(F\) of degree \(n\).

1. If \(n\) is prime to 2, prove that \(F(\alpha ^2)=F(\alpha )\).
2. Give an example in which \(n\) is prime to 3 but \(F(\alpha ^3)\ne F(\alpha )\).