Exercise 5, § 9

a) Let $K$ be a field, $\Omega$ an algebraic closure of $K$ and $E, F$ two subextensions of $\Omega$. Show that every composite extension of $E$ and $F$ (V, p. 12) is isomorphic to a composite extension of the form $(L_w, 1, w)$, where $w$ denotes a $K$-isomorphism of $F$ onto a subfield of $\Omega$, $1$ is the identity mapping of $E$ and $L_w$ is the subfield of $\Omega$ generated by $E \cup w(F)$. b) Deduce from a) that if $F$ is of finite degree $n$ over $K$, there are at most $n$ composite extensions of $E$ and $F$ that are pairwise non-isomorphic. c) Suppose that $E$ is a quasi-Galois extension of $K$ and $F$ a subextension of $E$. Show that every composite extension of $E$ and $F$ is isomorphic to a composite extension of the form $(E, 1, v)$, where $v$ is a $K$-isomorphism of $F$ onto a subfield of $E$.