Corollary (« Exchange Theorem »)
Tag 06S4, permanent. Cite it as [Bourbaki, Tag 06S4].
Algebra, chapter V, § 14. Transcendental extensions
— Let E be an extension of K, S a subset of E such that E is algebraic over K(S), T a subset of E which is algebraically free over K; then there exists a subset S' of S such that T ∪ S' is a transcendence basis of E over K and T ∩ S' = ∅.
For E is algebraic over K(T U S) and we have T ⊂ T U S.
Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.
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