nLab normed field

Idea

A normed field $K$ 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 $f,g \in K$ we have the equality ${\vert f \cdot g\vert} = {\vert f\vert} \cdot {\vert g\vert}$ instead of just the inequality ${\vert f \cdot g\vert} \leq {\vert f\vert} \cdot {\vert g\vert}$) 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

• Every field carries the trivial norm (which is non-archimedean), whose value is always $1$ (except that the norm of $0$ is $0$) and is complete with respect to this norm. (In constructive mathematics, either the field must be a discrete field or the norm must be allowed to take values in the lower real numbers.)

• The field $\mathbb{R}$ of real numbers and the field $\mathbb{C}$ of complex numbers, with their usual absolute value as the norm, are complete archimedian normed fields.

• For each prime number $p$, the field $\mathbb{Q}_p$ of $p$-adic numbers is a complete non-archimedean normed field with respect to the p-adic valuation.

• The field $\mathbb{Q}$ of rational numbers, with any of the norms in the two previous entries, is an incomplete normed field whose completion is $\mathbb{R}$ or $\mathbb{Q}_p$.

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

Related concepts

• normed division algebra

algebraic structure	group	ring	field	vector space	algebra
(submultiplicative) norm	normed group	normed ring	normed field	normed vector space	normed algebra
multiplicative norm (absolute value/valuation)			valued field
completeness	complete normed group	Banach ring	complete field	Banach vector space	Banach 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 $p$-adic Analysis and Applications, ICTP, Trieste, 31 August - 11 September 2009 (pdf)