ALGEBRA PRELIM August, 2000

If \(G\) is a group of order 105, prove that \(G\) has a normal cyclic subgroup of order 35.

Let \(f(x)\) be a polynomial of degree \(n>2\) over \(\mathbb {Q}\), the field of rational numbers. Let \(K\) be the splitting field of \(f(x)\) over \(\mathbb {Q}\) and suppose that the Galois group of \(f(x)\) over \(\mathbb {Q}\) is \(S_n\), the group of all permutations of \(n\) objects. Prove the following:

1. \(f(x)\) is irreducible over \(\mathbb {Q}\).
2. Let \(\alpha \) be a root of \(f(x)\) in \(K\) and let \(\sigma \) be an automorphism of the field \(\mathbb {Q}(\alpha )\). Prove that \(\sigma =1\).

Find an integer \(n\) such that the abelian group \(\mathbb {Z}/2\mathbb {Z}\oplus \mathbb {Z}/4\mathbb {Z}\oplus \mathbb {Z}/8\mathbb {Z}\) is a quotient of the multiplicative group \((\mathbb {Z}/n\mathbb {Z})^*\) of integers relatively prime to \(n\) modulo \(n\).

Let \(G\) be the Galois group of the polynomial \(x^3+2x-2\) 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 3.

Let \(H\) be a proper subgroup of the finite group \(G\). Prove that there is an element \(x\) in \(G\) such that \(x\) is not conjugate in \(G\) to any element of \(H\).

Let \(A\) and \(B\) be commuting \(n\times n\) matrices over a field \(F\). If \(A\) and \(B\) are both diagonalizable over \(F\), prove that \(A\) and \(B\) are simultaneously diagonalizable (i.e., there is an \(n\times n\) nonsingular matrix \(S\) over \(F\) such that both \(S^{-1}AS\) and \(S^{-1}BS\) are diagonal).

Let \(R\) be an infinite simple ring (i.e., a ring with infinitely many elements and with no 2-sided ideals except \(\{0\}\) and \(R\)). Show that exactly one of the following holds.

1. \(R\) is a division ring.
2. \(R\) has an infinite number of right ideals.

Let \(G\) be a nonabelian group of order \(p^3\) where \(p\) is a prime. Prove that \(Z(G)\) and \(G'\) are equal and have order \(p\). [Here \(Z(G)\) is the center of \(G\) and \(G'\) is the commutator subgroup of \(G\), i.e., \(G'\) is generated by all elements of the form \(x^{-1}y^{-1}xy\) with \(x,y\in G\).]

Determine all rings \(R\) with an identity such that \(R\) contains at most 9 elements and \(x^9=x\) for all \(x\in R\).

Suppose \(F\) is a field and \(K=F(\alpha )\) is a finite Galois extension of \(F\). Assume that the Galois group of \(K\) over \(F\) is cyclic generated by an automorphism \(\sigma \) where \(\alpha ^\sigma =\alpha +1\). Prove that \(F\) has characteristic \(p\ne 0\) and that \(\alpha ^p-\alpha \in F\).