Decide whether the following two polynomials are irreducible over : and .
Note that if a cubic polynomial in is reducible over , then it must have a linear factor- that is, it has a root in . By the rational root theorem, there are finitely many candidates for this root. If none of these is a root, then the polynomial is irreducible in .
By the rational root theorem, if has a root in it is in the set . Since , , and , is irreducible in .
Again by the rational root theorem, if has a root in then it is in the set . It is easy but rather tedious to see that , , , , , , , , , , , and . Thus is irreducible in .