Exercise 1, § 15

Let K be a field of characteristic p > 0, E an extension of K and B a p-basis of E over $K(E^p)$ (V, p. 98). a) Assume that $E \subset K^{p^{-1}}$; let $K_0$ be a subfield of K such that E is separable over $K_0$. Show that the set $B^p$ is a p-independent subset over $K_0(K^p)$ (observe that if $(a_\lambda)$ is a basis of the vector $K_0$-space $K$, then $(a_\lambda^p)$ is a basis of $K_0(K^p)$ over $K_0$. Let $C$ be a subset of $K$, disjoint from $B^p$ and such that $B^p \cup C$ is a p-basis of $K$ over $K_0(K^p)$; show that $B \cup C$ is a p-basis of $E$ over $K_0(E^p)$.

b) Assume that $E \subset K^{p^{-n}}$ and that $E$ is separable over a subfield $K_0$ of $K$. Show that if the degree of imperfection of $K$ over $K_n$ is finite (V, p. 170. Ex. 1), then it is equal to the degree of imperfection of $E$ over $K_0$ (use a)).