University of Utah, Department of Mathematics January 2014, Algebra Qualifying Exam

Let \(G\) be a group of order \(p^3\), where \(p\) is a prime. Suppose \(G\) is not abelian. Show that the center \(Z\) of \(G\) is isomorphic to \(\mathbb {Z}/(p)\) and that \(G/Z\cong \mathbb {Z}/(p)\times \mathbb {Z}/(p)\).

What is the number of elements of order 11 in a simple group of order 660?

Let \(M\) be the cokernel of the map \(\mathbb {Z}^2\longrightarrow \mathbb {Z}^3\) given by \[ \begin {pmatrix} 3 & 6\\ 4 & 10\\ 10 & 22 \end {pmatrix}. \] Write \(M\) as a direct sum of cyclic groups.

Let \(M\) be an \(n\times n\) matrix with entries from a field \(\mathbb {F}\) such that \(M^3=I\). Is \(M\) necessarily diagonalizable if (a) \(\mathbb {F}=\mathbb {Q}(\sqrt {3})\), (b) \(\mathbb {F}=\mathbb {Q}(i)\), (c) \(\mathbb {F}=\mathbb {F}_3\)?

Determine all \(3\times 3\) matrices \(M\) with entries from \(\mathbb {Q}\) such that \(M^8=I\) and \(M^4\ne I\).

Find all positive integers \(n\) such that \(\cos (2\pi /n)\) is rational.

Is the polynomial \(x^8+x+1\) irreducible in \(\mathbb {F}_2[x]\)?

Which of the following is a principal ideal domain? (a) \(\mathbb {Z}[i]\), (b) \(\mathbb {Z}[2\sqrt {2}]\), (c) \(\mathbb {Q}[3\sqrt {3}]\).

Let \(\mathbb {F}\) be a field with \(\mathbb {Q}\subset \mathbb {F}\subset \mathbb {C}\) such that \([\mathbb {F}:\mathbb {Q}]\) is odd.

1. If \(\mathbb {F}/\mathbb {Q}\) is Galois, prove that \(\mathbb {F}\) is contained in \(\mathbb {R}\).
2. Find an extension with \([\mathbb {F}:\mathbb {Q}]=3\) such that \(\mathbb {F}\) is not contained in \(\mathbb {R}\).

Prove that each element of a finite field can be written as a sum of two squares.