Exercise 2, § 13

a) Let $E$ be an extension of a field $K$ of characteristic $p > 0$ and $F$ an extension of $E$. Show that if $B$ is a $p$-basis of $E$ over $K(E^p)$ and $C$ a $p$-basis of $F$ over $E(F^p)$, then there exists a $p$-basis of $F$ over $K(F^p)$ contained in $B \cup C$.

b) Suppose that B is finite and that F is an algebraic extension of K of finite degree. Show that $[E : K(E^p)] \geq [F : K(F^p)]$. If moreover F is separable over E, show that $[E : K(E^p)] = [F : K(F^p)]$.