I think I might have an idea:

Continuing on in the same thinking to get my contradiction / show that , since is separable, that means its minimal polynomial, say factors into simple roots. Since , is irreducible in . Hence, . But that means . Thus .

The only fuzzy part that I cannot get past in this reasoning is the relationship between the minimal polynomial of an extension field (in this case the extension over containing ) and any irreducible polynomial in the base field. EDIT: Oh, and if is separable in then is it still separable in any extension field with as a root of some irreducible polynomial (not necessarily intermediate between base field and splitting field of the irreducible polynomial?...I mean...I don't think is an intermediate field of and the splitting field of ). I'll try to convince myself this answer is correct...