Student solutions for Assignment #3

Solution Sample

Problem 50

• Problem 50 - Tan Dang

Problem 94

Show $ f(x) = x^4 + 5x^2 + 3x + 2 $ is irreducible over the field of rational numbers.

• Solution by Nicole_Rutt

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] $

• 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

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