Proposition 13

— Let E be an algebraic extension of K and let E_s be the relative separable algebraic closure of K in E. a) E is a p-radical extension of E,. b) If F is a subextension of E such that E is p-radical over F, then F ⊃ E,. c) E, is the unique subextension of E which is separable over K and over which E is p-radical. It suffices to prove a) in the case where K is of characteristic $p \neq 0$. Let x be an element of E and f its minimal polynomial over K. There exists an integer $m \geq 0$ such that f belongs to $K[X^{p^m}]$ but not to $K[X^{p^{m+1}}]$; in other words, we have $f(X) = g(X^{p^m})$ with $g \in K[X]$, $g \notin K[X^p]$. Since f is irreducible, the same is true of g, hence g is the minimal polynomial of $x^{p^m}$ over K. By V, p. 38, Prop. 4 and p. 39, Prop. 5 we thus have $x^{p^m} \in E_s$, so E is p-radical over E,.

Assume now the hypothesis b) and let $x \in E,$. Since x is separable over K, it is so also over F (V, p. 39, Cor. 2), but since E is p-radical over F, x is also p-radical over F, hence $x \in F$ (V, p. 39, Cor. 3).

Finally c) follows from a) and b) and Prop. 12.