Exercise 6, § 9

a) Let K be a field of characteristic exponent p, L an extension of K, Ω an algebraic closure of L and $\overline{K}$ the relative algebraic closure of K in R. Show that $\overline{K}$ and $L^{p^{-\infty}}$ are linearly disjoint over $E = K \cap L^{p^{-\infty}}$. (Let $(a_\lambda)$ be a basis of the vector space $L^{p^{-\infty}}$ over E ; consider a relation $\sum_{\lambda} c_{\lambda} a_{\lambda} = 0$, where $c_\lambda \in \overline{K}$ and the number of non-zero $c_{\lambda}$ is as small as possible, and apply an L-automorphism of $\Omega$ to this relation.) b) Deduce from a) that if E and F are two extensions of K such that E is algebraic over K and K is relatively algebraically closed in F, then any two composite extensions of E and F are isomorphic.