Exercise 6, § 15

Let $K$ be a field of characteristic $p > 0$, $E$ a separable extension of $K$ and $B$ ap-basis of $E$ over $K(E^p)$. a) Show that $B$ is algebraically free over $K$ (consider an algebraic relation of least degree between the elements of $B$ and put the degrees of the variables which occur in the form $kp + h$ with $0 \leq h \leq p - 1$). Deduce that if $\operatorname{tr. deg}_K E < +\infty$, then the degree of imperfection of $E$ over $K$ is at most equal to $\operatorname{tr. deg}_K E$. b) Show that $E$ is separable and relatively perfect over $K(B)$.