Corollary 2

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

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

Let $K'$ be a locally compact field and $K$ a closed subfield of $K'$ (neither necessarily commutative). If $K'$ is a left (resp. right) vector space of finite dimension $n$ over $K$ , then (2)

\operatorname{mod}_{K'}(x) = (\operatorname{mod}_K(x))^n \quad \text{for all } x \in K.

In general it is known that in a (left or right) vector space of finite dimension $n$ over $K$ , the homothety with ratio $x \in K$ has modulus equal to $(\mathrm{mod}_K(x))^n$ ; it suffices to apply this to $K'$ .

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

Cited by

Nothing.

Cites

Nothing.