ALGEBRA EXAM - AUGUST 2007

Let \(\mathbb {F}_q\) be a finite field of order \(q\). Compute the order of \(GL_n(\mathbb {F}_q)\), the group of invertible \(n\times n\) matrices with coefficients in \(\mathbb {F}_q\).

Let \(I\subseteq \mathbb {Z}[x]\) be an ideal consisting of all polynomials \(a_0+a_1x+a_2x^2+\cdots \) such that 8 divides \(a_0\), 4 divides \(a_1\) and 2 divides all other coefficients. Find a finite set of generators of the ideal \(I\).

Let \(M\) be a \(\mathbb {Z}\)-module consisting of \(3\times 1\)-matrics with integer entries. Let \(N\subseteq M\) be a \(\mathbb {Z}\)-submodule generated by the columns of the matrix \[ \begin {pmatrix} 2 & 7 & 3 & 4\\ 3 & 8 & 7 & 11\\ 4 & 10 & 10 & 16 \end {pmatrix}. \] Find a minimal presentation for the quotient \(M/N\).

Show that the ring \(\mathbb {Z}[\sqrt {-1}]\) is a euclidean domain.

Let \(T\) be a complex square matrix such that \(T^{2007}=I\), where \(I\) is the identity matrix. Show that \(T\) can be diagonalized.

Show that \(\mathbb {Q}\otimes _{\mathbb {Z}}\mathbb {Q}=\mathbb {Q}\) and \(\mathbb {Q}\otimes _{\mathbb {Z}}(\mathbb {Z}/n\mathbb {Z})=0\).

Let \(\zeta _8=e^{\frac {\pi }{4}i}\). Determine the Galois group of \(K=\mathbb {Q}(\zeta _8)\) and then determine all quadratic extensions of \(\mathbb {Q}\) contained in \(K\).

Compute the Galois group \(G(E/F)\) where \(E\) is the splitting field of the polynomial \(x^3-2\) and \(F\) is

1. \(\mathbb {Q}\)
2. \(\mathbb {R}\)
3. \(\mathbb {F}_5\)
4. \(\mathbb {F}_7\)

Let \(m\) be a maximal ideal in \(\mathbb {R}[x,y]\) containing \(x^2+y^2+1\). What is the quotient \(\mathbb {R}[x,y]/m\)? Justify your answer.

Let \(D_8\) be the group of symmetries of a square. Show that the 2 dimensional representation (as symmetries of the square) is irreducible by calculating the character table.