University of Utah, Department of Mathematics Spring 2011, Algebra Qualifying Exam

Let \(p\) be a prime. Show that an element in the symmetric group \(S_n\) has order \(p\) if and only if it is a product of commuting \(p\)-cycles. Show by an explicit example that this need not be the case if \(p\) is not prime.

Prove that the number of Sylow \(p\)-subgroups of \(GL_2(\mathbb {F}_p)\) is \(p+1\).

Let \(G\) be a \(p\)-group with \(|G|>p\). Show that

1. \(G\) has nontrivial center;
2. \(G\) has a normal subgroup of every order \(p^m<|G|\).

For the matrix \[ A= \begin {pmatrix} 1 & 0 & 0\\ 0 & 0 & -2\\ 0 & 1 & 3 \end {pmatrix}, \] find:

1. the rational canonical form over \(\mathbb {Q}\);
2. the Jordan canonical form over \(\mathbb {C}\).

Prove that the ring \(\mathbb {Z}[i]\) is a Euclidean domain.

Let \(G\) be a finite abelian group and \(H\) a subgroup of \(G\). Show that \(G\) has a subgroup isomorphic with \(G/H\).

Let \(m,n\) be positive integers and \(d\) their greatest common divisor. Show that \[ \mathbb {Z}/m\mathbb {Z}\otimes _{\mathbb {Z}}\mathbb {Z}/n\mathbb {Z}\cong \mathbb {Z}/d\mathbb {Z}. \]

For a ring \(R\) define its nilradical \(n(R)=\{x\in R:x^n=0\text { for some }n\in \mathbb {Z}\}\).

1. If \(R\) is commutative, prove that \(n(R)\) is an ideal of \(R\).
2. Is the nilradical an ideal even if \(R\) is noncommutative?

What is the Galois group of \(x^4-5\) over:

1. \(\mathbb {Q}\);
2. \(\mathbb {Q}(\sqrt {5})\);
3. \(\mathbb {Q}(i)\).

Show that the extension \(\mathbb {Q}(\sqrt {2},\sqrt {3})\) is Galois over \(\mathbb {Q}\), and determine the Galois group.

Show that the polynomial \(x^2+y^2-1\) is irreducible in \(\mathbb {Q}[x,y]\). Is it irreducible in \(\mathbb {C}[x,y]\)?