Find a basis for over . Describe all of the bases for in terms of this basis.
Note that is a root of , which is irreducible over by Eisenstein’s criterion. In particular, is the minimal polynomial for over . As we saw in Theorem 4.6 in TAN, every element of is uniquely of the form , where . In particular, is a basis for over .
Suppose , , and is a subset of , and define on by , , and and extending linearly. Now is a basis of if and only if is a -isomorphism. Evidently, the matrix of with respect to the basis (in both domain and codomain) is . In turn, is an isomorphism if and only if is invertible, which holds if and only if is nonzero.