Math 551, Assignment 5, due Friday, November 1 in class

1. Let p be a prime and n a positive integer. Show that the number of isomorphism types of abelian groups of order p^{n} is equal to the number of conjugacy classes in Σ_{n} (i.e., the number of orbits in the action of Σ_{n} on itself by conjugation).

2. Show that if G, H, and K are finitely generated modules over a PID R, then G⊕H ≅ G ⊕ K implies H ≅ K. Give a counterexample if the finite generation assumption is removed.

3. Show that in the ring R = Z[\sqrt{−5}] := \{a + b\sqrt{−5} | a, b ∈ Z\}, 3 is irreducible but not prime.

4. Show that in a Euclidean domain R with rank function ν, an element r ∈ R is a unit if and only if r ≠ 0 and ν(r) = ν(1).

5. Let R = Z[i] := Z ⊕ Zi, the ring of all complex numbers with integer real and imaginary parts. Show that ν(a + bi) = |a + bi|^{2} makes R a Euclidean ring. Determine which elements of R are units, and show that a prime integer p ∈ Z is prime in R if and only if p is not the sum of two squares in Z.

6. Let R be a PID and let A be an m×n matrix with coefficients in R. An i×i minor of A is the determinant det B of an i×i matrix obtained by erasing some rows and columns of A. Let Δ_{i}(A) be the g.c.d. of all the nonzero i × i minors of A (and Δ_{i}(A) = 0 if all i × i minors are 0). Also define Δ_{0}(A) = 1. Theorem I implies (as will be discussed in class) that there exist P ∈ GL_{m}(R) and Q ∈ GL_{n}(R), and elements m_{1}, …, m_{r} ∈ R, r = min(m, n), such that m_{1}|m_{2}| ⋯ |m_{r} and

P AQ = [ \begin{matrix}m_{1} & 0 & 0 & ⋯ \\ 0 & m_{2} & 0 & ⋯ \\ 0 & 0 & m_{3} & ⋯ \\ · & · & · & ⋯ \end{matrix} ] .

Show that Δ_{i−1}m_{i} ∼ Δ_{i}(A) for any i such that Δ_{i−1}(A) ≠ 0. (Here a ∼ b means Ra = Rb.)

7. If the finitely generated module M over the PID R satisfies M ≅ R/p_{1}^{n_{1}}R⊕⋯ R/p_{r}^{n_{r}}R, where p_{1}, …, p_{r} are primes in R and n_{1}, …, n_{r} are positive integers, call

p_{1}^{n_{1}}, …, p_{r}^{n_{r}}

the list of elementary divisors of M. (If M is the 0 module, then r = 0.) Thus by Theorem III the list of elementary divisors is determined up to associates, and up to reordering the list. Show that if N is a submodule of M, then lists of elementary divisors of M and N may be reordered so that the elementary divisors of N are p_{1}^{k_{1}}, …, p_{s}^{k_{s}} for some 0 ≤ s ≤ r and some integers 1 ≤ k_{i} ≤ n_{i}, i = 1, …, s. (Hint: it was shown in class that for any prime power p^{n}, dim_{R/pR}((p^{n}M[p])) is the number of elementary divisors which are divisible by p^{n+1}.) Show conversely that every such list of elementary divisors arises from some submodule N of M (not necessarily unique).

Prove the corresponding results for quotient modules of M. (Hint. Use p^{n}M/p^{n+1}M instead of p^{n}M[p].)