Exercise 6, § 13

Let K be a field of characteristic $p > 0$, E an algebraic extension of K of finite degree and N the quasi-Galois extension of K generated by E. a) Let E_r and E_s (resp. N_r and N_s) be the largest p-radical extension and the largest separable extension of K contained in E (resp. in N). Show that for $E = E_r(E_s)$ (in which case E is isomorphic to $E_s \otimes_K E_r$) it is necessary and sufficient that $E_r = N_r$. b) If the degree of imperfection of E over K is 1, then a necessary and sufficient condition for $E = E_r(E_s)$ is that the degree of imperfection of N over K be 1. Deduce that if N is a quasi-Galois extension of K, whose degree of imperfection over K is equal to 1, then $E = E_r(E_s)$ for every subextension E of N (cf. Ex. 2, b)). c) Conversely if N is a quasi-Galois extension of K of finite degree whose degree of imperfection over K is $> 1$, show that if $N_r \neq N$, then there exists $t \in N$ such that for the extension $E = K(t)$ of K we have $E \neq E_r(E_s)$ (if a, b are two elements of a p-basis of N over $K(N^p)$ and if $x \in N$ is separable over K and $x \notin K$, take $t = a + bx$).