Student solutions for Assignment #3

Solution Sample

Problem 50

• Problem 50 - Tan Dang

Problem 94

Show f(x) = x⁴ + 5x² + 3x + 2 is irreducible over the field of rational numbers.

• Solution by Nicole_Rutt

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] $

• 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

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.

Solution by Nathan Moses

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