## Contents

# Student solutions for Assignment #3

## Problem 50

## Problem 73

Show that if *p* is a prime such that there is an integer *b* with *p* = *b*^{2} + 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)

- My solution only uses the fact that

- I think this is a bit cleaner.
- I very much prefer this solution! -- Avi 11:08, 26 June 2013 (UTC)

- I think this is a bit cleaner.

## Problem 94

Show *f*(*x*) = *x*^{4} + 5*x*^{2} + 3*x* + 2 is irreducible over the field of rational numbers.

## Problem 101

(a) Show that *x*^{4} + *x*^{3} + *x*^{2} + *x* + 1 is irreducible in $ \mathbb{Z}_3[x] $.

(b) Show that *x*^{4} + 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.