WRITTEN QUALIFYING EXAM - Math 6310 / Summer 2025

Let \(p\) be a prime number.

1. Find all the abelian groups with \(p^6\) elements.
2. How many abelian groups are there with a million (i.e. \(10^6\)) elements?

Find all the maximal ideals in the following rings:

1. \[ \mathbb {Q}[x,y]/\langle xy-2,x^2-y^2\rangle \]
2. \[ \mathbb {R}[x,y]/\langle xy-2,x^2-y^2\rangle \]
3. \[ \mathbb {C}[x,y]/\langle xy-2,x^2-y^2\rangle \]

Suppose that \(A\) is an \(n \times n\) matrix with characteristic polynomial

Prove that if \(A^d=A^e\) for any pair \(d\ne e\) of integers, then \(A=I_n\). And no, you cannot just quote the theorem that a matrix of finite order is diagonalizable. We want you to demonstrate that you know why it is true.

Consider the maximal ideal \(\langle x,y\rangle \subset k[x,y]\) for a field \(k\).

1. Prove that the kernel of the natural map of \(k[x, y]\)-modules:

   \[ \langle x,y\rangle \otimes _{k[x,y]}\langle x,y\rangle \to \langle x,y\rangle ^2 \]

   is a one-dimensional vector space over \(k\) and find a generator of it.

2. Find the other two Tors:

   \[ \operatorname {Tor}^{k[x,y]}_1(\langle x,y\rangle ,\langle x,y\rangle )\quad \text {and}\quad \operatorname {Tor}^{k[x,y]}_2(\langle x,y\rangle ,\langle x,y\rangle ) \]

What about the Homomorphisms in Problem 4? Namely,

1. Is it true that:

   \[ \operatorname {Hom}_{k[x,y]}(\langle x,y\rangle ,\langle x,y\rangle )=k\cdot \operatorname {id} \]

   is a one-dimensional vector space, or are there more \(k[x, y]\)-module homomorphisms than just the multiples of the identity?

2. Are there any non-split extensions of \(k[x, y]\)-modules:

   \[ 0\to \langle x,y\rangle \to M\to \langle x,y\rangle \to 0 \]

   i.e. is

   \(\operatorname {Ext}^1_{k[x,y]}(\langle x,y\rangle ,\langle x,y\rangle )=0\) or not?