Suppose is algebraic over with minimal polynomial . ( is indeed irreducible since is Eisenstein at 2.) Compute the minimal polynomial of over .
Note that has degree at most 3 over . Evidently, we have and . Suppose ; comparing coefficients, we have the following system of linear equations: , , and . Evidently then is a root of . We can see using the rational root theorem that has no linear factors over , and thus is irreducible. So is the minimal polynomial of over .