@egreg You remember the problem we were trying to do, about atmost one non-separable element? Can you please give me your proof. Mine is broken, in the following sense: separable closure of F is a subfield of F. And, between the ground and the separable closure there are only finitely many extensions. And, then, there is a purely inseparable extension. Now, you inevitably assume characteristic p and then, note that, the minpoly of beta is of the form g(x^{q}) where q = p^e (and g separable).
And, finally, we'd have to do some more magic. So, I'd prefer your proof to mine if its constructiv…