Let be a field and let be a polynomial. Prove that is a field if and only if is irreducible.
Suppose is a field. Then is a maximal ideal and hence is prime. So is prime in ; since is a unique factorization domain, is irreducible.
Conversely, suppose is irreducible. Then is prime, so that is a prime ideal. Since is a principal ideal domain, is maximal, so that is a field.