Proposition 6

Tag 04SV, permanent. Cite it as [Bourbaki, Tag 04SV].

Commutative Algebra, chapter VI, § 2. Places

Let $K$ be a field and $S$ a subring of $K.$ For an element $x \in K$ to be integral over $S,$ it is necessary and sufficient that every place of $K$ which is finite on $S$ be finite at $x.$

This is also a consequence of Theorem 3 of § 1, no. 3.

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

Cited by

Proposition 5, § 3

Cites

Theorem 3 of § 1, no. 3