Exercise 12, § 14

a) Let K be a field, $\Omega$ an algebraically closed extension of K, $E \subset \Omega$ and F two extensions of K; assume that tr. $\deg_E \Omega \geq \operatorname{tr.} \deg_K F$. Show that every composite extension of E and F is isomorphic to a composite of the form $(L_w, 1, w)$, where w is a K-isomorphism of F onto a subfield of $\Omega$ and $L_w$ is the subfield of $\Omega$ generated by E and $w(F)$. b) Deduce from a) that if F is algebraic of finite degree n over K, then there are at most n composite extensions of E and F that are pairwise non-isomorphic.