The deriative of over is 1, which implies that the deriative has no root at all. Thus is separable. It follows that is separable over F, where is a root of p(x). Let . We see that is irreducible in . However, is factored as in . Thus E is purely inseparable over B.

[EDIT]show that there is no intermediate field B' such that E is separable over B' and B' purely inseparable over F.

Observe that , where and p is a prime number.Q2: Suppose that F is a field of characteristic p and that @ is a separable element in some extension. Show that then F(@^p)=F(@)

Any insights would be great!