Exercise 4, § 17

Let $K$ be a field and $E, F$ two extensions of $K$. a) Suppose that $K$ is relatively algebraically closed in $F$. Show that if $(L, u, v)$, $(L', u', v')$ are two composite extensions of $E$ and $F$ such that $u(E)$ and $v(F)$ (resp $u'(E)$ and $v'(F)$) are algebraically disjoint over $K$, then these two composite extensions are isomorphic. (Consider first the case where $E$ is a pure transcendental extension of $K$; then use Ex. 2, a), as well as Ex. 6, b) of V, p. 154) b) Let $E_s, F_s$ be the relative separable closures of $K$ in $E$ and $F$ respectively. Suppose that all the composite extensions of $E$, and $F_s$ are isomorphic. Show that if $(L, u, v)$ and $(L', u', v')$ are two composite extensions of $E$ and $F$ such that $u(E)$ and $v(F)$ (resp. $u'(E)$ and $v'(F)$) are algebraically disjoint over $K$, then these two composite extensions are isomorphic (use a) and Ex. 11 of V, p. 173).