ALGEBRA PRELIM August, 2003

Suppose \(H\) is a subgroup of the group \(G\) such that \(x^2\in H\) for all \(x\in G\). Prove that \(H\) is a normal subgroup of \(G\).

Suppose \(A\) is an \(n\times n\) complex matrix with \(n>1\). If \(A\) has rank 1, prove that \(A\) is similar to a diagonal matrix if, and only if, \(\operatorname {trace}(A)\ne 0\). [The trace of a matrix is the sum of the entries on the main diagonal.]

Let \(R\) be the ring of polynomials in one variable over a field \(F\). Prove that \(R\) has infinitely many maximal ideals.

1. Suppose \(K\) is a field of prime order and \(E\) is the algebraic closure of \(K\). If \(F\) is any subfield of \(E\), prove that the multiplicative group of \(F\) is periodic, i.e., every element of the multiplicative group has finite order.
2. Suppose \(F\) is a field whose multiplicative group is periodic. Prove that for some prime \(p\), \(F\) is isomorphic to a subfield of the algebraic closure of the field of order \(p\).

Let \(G\) be the Galois group of the polynomial \(x^3-3x^2-3\) over the field \(F\). Determine \(G\) in each of the following cases:

1. \(F\) is the complex numbers.
2. \(F\) is the real numbers.
3. \(F\) is the rational numbers.
4. \(F\) is the integers modulo 2.

If \(G\) is a simple group of order 60, prove that \(G\) is isomorphic to \(A_5\), the alternating group of degree 5.

Suppose \(G\) is a group of order \(p^n\) where \(p\) is a prime and \(n>1\). If \(A\) is the group of all automorphisms of \(G\), prove that \(p\) divides the order of \(A\).

Determine, with proof, all rings \(R\) such that \(R\) contains at most 11 elements and \(x^{11}=x\) for all \(x\in R\).

Prove that there is a rational polynomial whose Galois group over the rationals is the direct product of two groups of order 3.