Let be a root of . Show that is irreducible in .
First we will verify that is irreducible over . To that end, since has degree 3, it suffices to show that it has no linear factor. Via the rational root theorem, it then suffices to note that while . So is the minimal polynomial for over .
Note that is the field polynomial for over . Thus over this field; since the norm of is a rational prime, is irreducible as an algebraic integer in .