Exercise 2, § 15

Let $K$ be a field of characteristic $p > 0$, $E$ an extension of $K$ and $F$ a separable extension of $E$. Show that any two of the following three properties imply the third : a) $B$ is a p-basis of $E$ over $K(E^p)$; b) $C$ is a p-basis of $F$ over $E(F^p)$; c) $B \subset E,\ B \cup C$ is a p-basis of $F$ over $K(F^p)$ and $B \cap C = \varnothing$. (Use the fact that if $(c_\mu)$ is a basis of the vector E-space $F$, then $(c_\mu^p)$ is a basis of $K(E^p)[F^p]$ over $K(E^p)$ and of $E[F^p]$ over $E$.)