Proposition 2

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

Commutative Algebra, chapter VI, § 9. Application: locally compact fields

Let $K, K'$ be two (not necessarily commutative) locally compact fields such that $K$ is a topological subfield of $K'$ and $K$ is not discrete. Then: (i) $K'$ is a finite dimensional left (resp. right) vector space over $K$ . (ii) If $K$ is contained in the centre of $K'$ , then, for all $x \in K'$ . (1)

\operatorname{mod}_{K'}(x) = \operatorname{mod}_K(N_{K'/K}(x)).

K

is a complete valued field which is not discrete, assertion (i) follows from Topological Vector Spaces, Chapter I, § 2, no. 4, Theorem 3; assertion (ii) is just Integration, Chapter VII, § 1, no. 11, Proposition 17.

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

Cited by

Theorem 1, § 9

Cites

Integration, Chapter VII, § 1, no. 11, Proposition 17 not yet on this site: INT VII
Topological Vector Spaces, Chapter I, § 2, no. 4, Theorem 3 not yet on this site: EVT I