Proposition 17

— Let E be an extension of a field K and B a transcendence basis of E over K. For E to be finitely generated over K (V, p. 11, Def. 2) it is necessary and sufficient that B be finite and that the degree $[E : K(B)]$ be finite.

Suppose that E is finitely generated over K and let S be a finite subset of E such that $E = K(S)$. By Th. 2 (V, p. 109) S contains a transcendence basis $B'$ of E over K and this has the same cardinal as B (V, p. 110, Th. 3). Therefore B is finite. Put $K' = K(B)$; then E is algebraic over $K'$ and we have $E = K'(S)$; since S is finite, Th. 2 of V, p. 18 shows that $[E : K']$ is finite.

Conversely, suppose that B is finite and $[E : K(B)]$ is finite. If C is a (finite) basis of the vector space E over $K(B)$, we have $E = K(B)(C) = K(B \cup C)$ and so E is a finitely generated extension of K.