University of Utah, Department of Mathematics January 2013, Algebra Qualifying Exam

Determine the number of Sylow \(p\)-subgroups of \(GL_2(\mathbb {F}_p)\).

Show that \((\mathbb {Q}/\mathbb {Z},+)\) has one and only one subgroup of order \(n\), for each integer \(n\ge 1\), and that this subgroup is cyclic.

Determine representatives for the conjugacy classes in \(GL_3(\mathbb {F}_2)\).

Let \(R\) be a commutative ring with \(1\ne 0\). Recall that the nilradical of \(R\) is the ideal \[ \mathcal {N}(R)=\{x\in R:x^n=0\text { for some positive integer }n\}. \] Prove that the following are equivalent:

1. \(R\) has exactly one prime ideal;
2. \(R/\mathcal {N}(R)\) is a field.

If \(I,J\) are ideals in the commutative ring \(R\), prove that \[ R/I\otimes _R R/J\cong R/(I+J), \] as \(R\)-modules.

In the category of \(\mathbb {Z}\)-modules:

1. Is \(\mathbb {Z}\) injective?
2. Is \(\mathbb {Z}/8\mathbb {Z}\) projective?

Show that if \(p\) is an odd prime, the polynomial \(x^{p^n}-x+1\) is irreducible over \(\mathbb {F}_p\) only when \(n=1\).

Show that \(f(x)=x^4+4x^2+2\) is irreducible over \(\mathbb {Q}\), and find its Galois group over \(\mathbb {Q}\).

Let \(f(x)\in \mathbb {Q}[x]\) be a polynomial of degree \(n\ge 4\) and let \(K\) be a splitting field of \(f\) over \(\mathbb {Q}\). Suppose that \(\operatorname {Gal}(K/\mathbb {Q})\) is the symmetric group \(S_n\). If \(\alpha \in K\) is a root of \(f(x)\), show that \(\alpha ^n\notin \mathbb {Q}\).

Let \(A\) be a real \(n\times n\) matrix. We say that \(A\) is a difference of two squares if there exist real \(n\times n\) matrices \(B\) and \(C\) with \(BC=CB=0\) and \(A=B^2-C^2\).

1. If \(A\) is a diagonal matrix, show that it is a difference of two squares.
2. If \(A\) is a symmetric matrix that is not necessarily diagonal, again show that it is a difference of two squares.
3. Suppose \(A\) is a difference of two squares, with corresponding matrices \(B\) and \(C\) as above. If \(B\) has a nonzero real eigenvalue, prove that \(A\) has a positive real eigenvalue.