Exercise 2, § 17

Let K be a field of characteristic p > 0, $\Omega$ an extension of K and E, F two subextensions of $\Omega$, algebraically disjoint over K. a) Let F_s be the relative separable closure of K in F. Show that E(F_s) is the relative separable closure of E in E(F). (Reduce to the case where F_s = K; let B be a transcendence basis of E over K and K' the relative algebraic closure of K in F, which is radical over F; then (V, p. 137) K'(B) is the relative algebraic closure of K(B) in F(B) and is p-radical over K(B). If x ∈ E(F) is algebraic over E, there exists an integer r ≥ 0 such that $x^{p^r} \in M$, where M is a separable extension of finite degree of F(B) having a basis $(u_i)_{1 \leq i \leq n}$ over $F(B)$ consisting of elements of $E$. If $y = \sum b_i u_i$ with $b_i \in F(B)$, show by considering the $F(B)$-isomorphisms of $M$ into an algebraic closure of $E(F)$ that the $b_i$ belong to $K'(B)$ and deduce that $y$ is p-radical over $E$.) b) Suppose that $E$ and $F$ are linearly disjoint over $K$, that $K$ is relatively algebraically closed in $F$ and that $E$ is a separable extension of $K$. Show that $E$ is relatively algebraically closed in $E(F)$. (Reduce to the case where $E$ is a separable algebraic extension of $K$, with the help of V, p. 138. Deduce from a) that the relative algebraic closure of $E$ in $E(F)$ is p-radical over $E$ and then use the fact that if $(a_i)$ is a basis of $E$ over $K$, the $a_i^{p'}$ also form a basis of $E$ over $K$.)