Exercise 5, § 14

Let $\Omega$ be an algebraically closed extension of a field K. Show that for every subextension E of $\Omega$, which is transcendental over K, there exist an infinity of K-isomorphisms of E onto a subfield of $\Omega$ (if x $\in$ E is transcendental over K, consider a subextension F of E such that x is transcendental over F and E algebraic over F(x), and show that there exist an infinity of F-isomorphisms of E onto a subfield of $\Omega$).