Corollary 1

— Suppose that E is a finitely generated extension of K, and let $K'$ be the relative algebraic closure of K in E (V, p. 19). Then $K'$ is of finite degree over K.

Let B be a transcendence basis of E over K. By Cor. 2 of V, p. 114, $K'$ is linearly disjoint from $K(B)$ over K, whence $[K' : K] = [K'(B) : K(B)] \leq [E : K(B)]$ and now the finiteness of $[K' : K]$ follows from that of $[E : K(B)]$.