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== Problem 101 == | == Problem 101 == | ||
| − | (a) Show that | + | (a) Show that <math>x^4 +x^3 +x^2 +x+1</math> is irreducible in <math>\mathbb{Z}_3[x]</math>. |
| − | (b) Show that | + | (b) Show that <math>x^4 + 1</math> is not irreducible in <math>\mathbb{Z}_3[x]</math> |
*[[Media:Week_3_Problem_101.pdf| Solution]] | *[[Media:Week_3_Problem_101.pdf| Solution]] | ||
Revision as of 06:17, 25 June 2013
Contents
Student solutions for Assignment #3
Problem 50
Problem 94
Show $ f(x) = x^4 + 5x^2 + 3x + 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 $
