(→Problem 101: fixed latex) |
|||
| Line 2: | Line 2: | ||
= Student solutions for Assignment #3 = | = Student solutions for Assignment #3 = | ||
| − | [[ | + | |
| + | [[Solution sample|Solution Sample]] | ||
| + | |||
---- | ---- | ||
| + | |||
== Problem 50 == | == Problem 50 == | ||
| − | *[[Media:Prob50.pdf|Problem 50 - Tan Dang]] | + | |
| + | *[[Media:Prob50.pdf|Problem 50 - Tan Dang]] | ||
== Problem 94 == | == Problem 94 == | ||
| − | Show < | + | |
| − | *[[Media:Problem_94_-_Nicole_Rutt.pdf| Solution by Nicole_Rutt]] | + | 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 == | == 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>. |
| − | *[[Media:Week_3_Problem_101.pdf| Solution]] | + | (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 == | == Problem 107 == | ||
| − | |||
| − | *[[Assn3Prob107|Solution by Avi Steiner]] | + | 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 kiwi.ecn.purdue.edu/rhea/images/c/ca/Problem_114_Kent_State_Algebra_Qual_Ring.pdf]<br> | ||
---- | ---- | ||
Revision as of 14:18, 25 June 2013
Contents
Student solutions for Assignment #3
Problem 50
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.
kiwi.ecn.purdue.edu/rhea/images/c/ca/Problem_114_Kent_State_Algebra_Qual_Ring.pdf
