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| − | <br> = Student solutions for Assignment #3 = [[Solution sample|Solution Sample]] ---- == Problem 50 == *[[Media:Prob50.pdf|Problem 50 - Tan Dang]] == 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. *[[ | + | <br> |
| + | |||
| + | = Student solutions for Assignment #3 = | ||
| + | |||
| + | [[Solution sample|Solution Sample]] | ||
| + | |||
| + | ---- | ||
| + | |||
| + | == Problem 50 == | ||
| + | |||
| + | *[[Media:Prob50.pdf|Problem 50 - Tan Dang]] | ||
| + | |||
| + | == 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. | ||
| + | |||
| + | *[[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]] | ||
| + | |||
| + | ::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) | ||
| + | |||
| + | *[[MA553QualStudyAssignment3Problem73Solution|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 <span class="texhtml">''f''(''x'') = ''x''<sup>4</sup> + 5''x''<sup>2</sup> + 3''x'' + 2</span> is irreducible over the field of rational numbers. | ||
| + | |||
| + | *[[Media:Problem_94_-_Nicole_Rutt.pdf|Solution by Nicole_Rutt]] | ||
| + | |||
| + | == Problem 101 == | ||
| + | |||
| + | (a) Show that <span class="texhtml">''x''<sup>4</sup> + ''x''<sup>3</sup> + ''x''<sup>2</sup> + ''x'' + 1</span> is irreducible in <math>\mathbb{Z}_3[x]</math>. | ||
| + | |||
| + | (b) Show that <span class="texhtml">''x''<sup>4</sup> + 1</span> is not irreducible in <math>\mathbb{Z}_3[x]</math> | ||
| + | |||
| + | *[[Media:Week_3_Problem_101.pdf|Solution]] | ||
| + | |||
| + | == Problem 107 == | ||
| + | |||
| + | Let <span class="texhtml">''R''</span> be a commutative ring with identity such that the identity map is the only ring automorphism of <span class="texhtml">''R''</span>. Prove that the set <span class="texhtml">''N''</span> of all nilpotent elements of <span class="texhtml">''R''</span> is an ideal of <span class="texhtml">''R''</span> | ||
| + | |||
| + | *[[Assn3Prob107|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. | ||
| + | |||
| + | *[https://kiwi.ecn.purdue.edu/rhea/images/c/ca/Problem_114_Kent_State_Algebra_Qual_Ring.pdf Solution by Nathan Moses]<br> | ||
| + | |||
| + | ---- | ||
| + | |||
| + | [[2013 Summer MA 598A Weigel|Back to 2013 Summer MA 598A Weigel]] | ||
| + | |||
| + | [[Category:MA598ASummer2013Weigel]] [[Category:Math]] [[Category:MA598]] [[Category:Problem_solving]] [[Category:Algebra]] | ||
Latest revision as of 06:11, 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)
- 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) = 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.
