# Proofs with sets

(See attached file for full problem description).

a) Determine the irreducibility of x20-11 over Q(set of rationals), and use it to prove or disprove that the ideal <x20-11> is a maximal ideal of Q[x].

b) Construct an integral domain R and an element a in R such that a is irreducible but not prime in R.

c) Suppose that R is a principal ideal domain and a in R is irreducible. If a does not divide b in R, prove that a and b are relatively prime.

d) Suppose p in N(set of naturals) is a prime number. Show that every element a in Zp has a p-th root, i.e. there is b in Zp with a=bp.

https://brainmass.com/math/combinatorics/proofs-sets-77354

#### Solution Summary

This solution shows how to determine irreducibility and use it in a proof of ideal, construct an integral domain and element with given characteristics, complete a proof that a and b are relatively prime, and complete a proof regarding roots of elements in a group.