# Student solutions for Assignment #3

## Problem 73

Show that if p is a prime such that there is an integer b with p = b2 + 4, then $\mathbb{Z}[\sqrt{p}]$ is not a unique factorization domain.

Here's my alternate proof - I found a few things that need to be changed in Avi's, which I can discuss in class.
My solution only uses the fact that p is a sum of two squares (i.e. is congruent to 1 mod 4), so I'm not sure it's correct. -- Avi 20:05, 25 June 2013 (UTC)
I think this is a bit cleaner.
I very much prefer this solution! -- Avi 11:08, 26 June 2013 (UTC)

## Problem 94

Show f(x) = x4 + 5x2 + 3x + 2 is irreducible over the field of rational numbers.

## Problem 101

(a) Show that x4 + x3 + x2 + x + 1 is irreducible in $\mathbb{Z}_3[x]$.

(b) Show that x4 + 1 is not irreducible in $\mathbb{Z}_3[x]$

## Problem 107

Let R be a commutative ring with identity such that the identity map is the only ring automorphism of R. Prove that the set N of all nilpotent elements of R is an ideal of R

## Problem 114

A local ring is a commutative ring with 1 that has a unique maximal ideal. Show that a ring R is local if and only if the set of non-units in R is an ideal.