Exercise 16, § 14

a) Let K be an algebraically closed field, $\Omega$ an algebraically closed extension of K, E a subextension of $\Omega$ and x an element of $\Omega$ transcendental over E. Let $\overline{K(x)}$ be the relative algebraic closure of $K(x)$ in $\Omega$ and let $M = E(\overline{K(x)})$; show that M is an algebraic extension of infinite degree of $E(x)$. (Observe that for every integer $m > 1$ the polynomial $X^m - x$ is irreducible in $E(x)[X]$ (V, p. 91, Remark). b) Let E be an algebraically closed field and F a finitely generated extension of E. Show that every algebraically closed subfield of F is necessarily contained in E. (Otherwise there would exist a subfield $L \subset F$ which is algebraically closed and an element $x \in L$ which is transcendental over E. Apply a), taking for K the relative algebraic closure in E of the prime subfield of E, noting that M should be finitely generated over E.)