Student solutions for Assignment #3

Solution Sample

Problem 50

• Problem 50 - Tan Dang

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.

• Solution by Andrew Zeller

Here's my alternate proof - I found a few things that need to be changed in Avi's, which I can discuss in class.

• Solution by Avi Steiner

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)

• Solution by Ryan Spitler

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.

• Solution by Nicole_Rutt

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] $

• Solution

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

• Solution by Avi Steiner

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.

• Solution by Nathan Moses
  

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