ALGEBRA EXAM - JANUARY 2008

Show that a finite group of order 105 has a non-trivial normal subgroup.

Let \(p\) be a prime. Determine all groups of order \(p^2\).

Let \(R\) be a commutative ring with 1. Assume that \(R\) satisfies the ascending chain condition. Let \(I\) be an ideal generated by an infinite sequence of elements \(x_1,x_2,\ldots \) in \(R\). Show that \(I\) is finitely generated.

Let \(\omega =-\frac {1}{2}+\frac {\sqrt {-3}}{2}\), a cube root of 1. Show that the ring \(\mathbb {Z}[\sqrt {\omega }]\) is a euclidean domain.

Let \(A\) be a rational \(3\times 3\) matrix such that \(A^3=A\). Show that \(A\) can be diagonalized.

Describe all finitely generated \(\mathbb {Z}\)-submodules of \(\mathbb {Q}\).

Show that \(1\otimes (1,1,\ldots )\ne 0\) in the tensor product \(\mathbb {Q}\otimes _{\mathbb {Z}}\prod _{n=2}^{\infty }(\mathbb {Z}/n\mathbb {Z})\).

Prove that \(\Phi _{2^n}(x)=x^{2^{n-1}}+1\) is irreducible in \(\mathbb {Q}[x]\).

Find a real number \(\alpha \) such that \(\mathbb {Q}(\alpha )\) is a Galois extension of \(\mathbb {Q}\) with the Galois group \(\mathbb {Z}/5\mathbb {Z}\).

Let \(G\) be the group of symmetries of an equilateral triangle. Show that the 2 dimensional representation (as symmetries of the triangle) is irreducible by calculating the character table.