For a field F, characterize the quotients of F[x] which are fields

Let F be a field and let p(x) \in F[x] be a polynomial. Prove that F[x]/(p(x)) is a field if and only if p(x) is irreducible.

Suppose F[x]/(p(x)) is a field. Then (p(x)) is a maximal ideal and hence is prime. So p(x) is prime in F[x]; since F[x] is a unique factorization domain, p(x) is irreducible.

Conversely, suppose p(x) is irreducible. Then p(x) is prime, so that (p(x)) is a prime ideal. Since F[x] is a principal ideal domain, (p(x)) is maximal, so that F[x]/(p(x)) is a field.

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