Show that and are irreducible over .
Recall that the Gaussian integers are a Unique Factorization Domain, and that . We claim that is the field of fractions of . Indeed, if is a field containing , then also contains any Gaussian rational since we can write with . By the universal property of fields of fractions, is the field of fractions of .
Recall that and 3 are irreducible in . So Eisenstein’s criterion applies, showing that and are irreducible over .