Corollary 3

— If E is a finitely generated extension of a field K, then every subextension E' of E is finitely generated.

Let B' be a transcendence basis of E' over K. By V, p. 109, Th. 2, B' is contained in a transcendence basis B of E over K and so is finite, by Prop. 17. Since E' is algebraic over K(B') and E is a finitely generated extension of K(B'). Cor. 1 shows $[E' : K(B')]$ to be finite. Now Prop. 17 shows that E' is finitely generated over K.

Prop. 17 may be paraphrased by saying that a finitely generated extension of K is an algebraic extension of finite degree of a purely transcendental extension $K(x_1, \ldots, x_n)$.