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== Problem 73 == | == Problem 73 == | ||
| − | Show that if <span class="texhtml">''p''</span> is a prime such that there is an integer <span class="texhtml">''b''</span> with <span class="texhtml">''p'' = ''b''<sup>2</sup> + 4</span>, then <math>\mathbb{Z}[\sqrt{p}]</math> is not a unique factorization domain. | + | Show that if <span class="texhtml">''p''</span> is a prime such that there is an integer <span class="texhtml">''b''</span> with <span class="texhtml">''p'' = ''b''<sup>2</sup> + 4</span>, then <math>\mathbb{Z}[\sqrt{p}]</math> is not a unique factorization domain. |
| + | |||
| + | *[[Media:Problem_73_Zeller.pdf|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. | ||
*[[Media:Prob_73.pdf|Solution by Avi Steiner]] | *[[Media:Prob_73.pdf|Solution by Avi Steiner]] | ||
Revision as of 02:42, 26 June 2013
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 = 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)
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.
