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# Proofs with sets

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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.

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#### 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.

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