nLab
normed field

Context

Analysis

Algebra

Contents

Idea

A normed field KK is a normed ring whose underlying ring is a field.

If the product preserves the norm strictly (so that one has a multiplicative norm or absolute value in that for all f,gKf,g \in K we have the equality |fg|=|f||g|{\vert f \cdot g\vert} = {\vert f\vert} \cdot {\vert g\vert} instead of just the inequality |fg||f||g|{\vert f \cdot g\vert} \leq {\vert f\vert} \cdot {\vert g\vert}) then one speaks of a valued field (e.g. Berkovich 09, def. 1.1.1).

If the underlying normed group is a complete topological space then one speaks of a complete normed field.

Examples

Properties

Relation to algebraic closure

The norm of a non-archimedean field extends uniquely to its algebraic closure and the completion of that with respect to this norm is still algebraically closed (Bosch-Guntzer-Remmert 84, prop. 3.4.1.3, Berkovich 09, fact 1.1.4).

For example the p-adic complex numbers p\mathbb{C}_p arise this way from the p-adic rational numbers p\mathbb{Q}_p.

algebraic structuregroupringfieldvector spacealgebra
(submultiplicative) normnormed groupnormed ringnormed fieldnormed vector spacenormed algebra
multiplicative norm (absolute value/valuation)valued field
completenesscomplete normed groupBanach ringcomplete fieldBanach vector spaceBanach algebra

References

  • Leonard Tornheim, Normed fields over the real and complex fields, Michigan Math. J. Volume 1, Issue 1 (1952), 61-68. (Euclid)

  • S. Bosch, U. Guntzer, and R. Remmert, Non-Archimedean analysis, Grundlehren der Mathematischen Wissenschaften, vol. 261, Springer-Verlag, Berlin, 1984.

  • Vladimir Berkovich, Non-archimedean analytic spaces, lectures at the Advanced School on pp-adic Analysis and Applications, ICTP, Trieste, 31 August - 11 September 2009 (pdf)

Last revised on July 13, 2014 at 07:58:57. See the history of this page for a list of all contributions to it.