Let be an irreducible cubic polynomial over . Suppose is an extension of of degree 4; prove that is irreducible over .
We may assume without loss of generality that is monic.
Now suppose to the contrary that is reducible over . Since is cubic, it must have a linear factor, and thus must have a root in . Since is a root of the monic irreducible over , is in fact the minimal polynomial of over , and thus . We thus have the following chain of fields: . By Theorem 5.7 in TAN, we have that 3 divides 4, a contradiction.
So is irreducible over .