Bourbaki

Corollary 2

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

Commutative Algebra, chapter VI, § 8. Extensions of a valuation to an algebraic extension

If K^\hat{K} or L is separable over K, the canonical mapping

ϕ:K^⊗KL→∏i=1sL^i \phi : \hat{K} \otimes_K L \to \prod_{i=1}^s \hat{L}_i

is an isomorphism.

The Jacobson radical of K^⊗KL\hat{K} \otimes_K L is then zero (Algebra, Chapter VIII, § 7, no. 3, Theorem 1).

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

Cited by

Nothing.

Cites

Transcribed from the printed volume. Not translated.