## 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.