Set F=F2(t) where t is a variable, let @ be a root of the equation x^2+x+t=0, set B=f(@) and set b=@^(1/2) and E=F(b). then B is separable over F and E is purely inseparable over B. show that there is no intermediate field B' such that E is separable over B' and B' purely inseparable over F.

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!