# nLab discrete valuation

There are two different notions of a discrete valuation.

A discrete valuation on a field $K$ is a function $v:K\to Z\cup \left\{\infty \right\}$ such that

• $v$ defines the homomorphism of groups $v{\mid }_{K}:{K}^{*}\to Z$ where ${K}^{*}$ is the multiplicative group of $K$

• $v\left(0\right)=\infty$

• $v\left(x+y\right)\ge \mathrm{inf}\left\{v\left(x\right),v\left(y\right)\right\}$

with usual conventions for $\infty$.

The set ${R}_{v}=\left\{x\in K\phantom{\rule{thinmathspace}{0ex}}\mid \phantom{\rule{thinmathspace}{0ex}}v\left(x\right)\ge 0\right\}$ is called the valuation ring of the valuation $v$, and it is an integral domain with quotient field $K$; its part ${𝔭}_{v}=\left\{x\in K\phantom{\rule{thinmathspace}{0ex}}\mid \phantom{\rule{thinmathspace}{0ex}}v\left(x\right)>0\right\}$ is a maximal ideal of ${R}_{v}$ and is called the valuation ideal of $v$. This can be generalized to other valuations.

Let $0<\rho <1$ and define $\mid x{\mid }_{v}={\rho }^{v\left(x\right)}$. Then $\mid \mid$ will be a nonarchimedean multiplicative discrete valuation in the sense of the following definition. If we change a $\rho$ then we get an equivalent multiplicative valuation.

A valuation on a field $K$ is a function $\mid \mid :K\to {R}_{\ge 0}$ such that

• $\mid x\mid =0$ iff $x=0$

• $\mid xy\mid =\mid x\mid \mid y\mid$

• there is a constant $C$ such that $\mid 1+x\mid \le C$ if $\mid x\mid \le 1$

Two valuations on the same field are equivalent if one is the positive power of another i.e. $\exists c>0$ such that $\mid x{\mid }_{1}=\mid x{\mid }_{2}^{c}$ for all $x\in K$.

A valuation is non-archimedean if $C$ above can be taken $1$ and archimedean otherwise. A (multiplicative) valuation is discrete if there is a neighborhood $U\ni 1\in {R}_{+}$ such that the only $x\in K$ such that $\mid x\mid \in U$ is ${1}_{K}\in K$.

By a theorem of Gel’fand and Tornheim the only archimedean valuation fields are the subfields of $C$ with a valuation which is equivalent to the valuation obtained by resriction from the standard absolute value on $C$.

• A. Fröhlich, J. W. S. Cassels (editors), Algebraic number theory, Acad. Press 1967, with many reprints; Fröhlich, Cassels, Birch, Atiyah, Wall, Gruenberg, Serre, Tate, Heilbronn, Rouqette, Kneser, Hasse, Swinerton-Dyer, Hoechsmann, systematic lecture notes from the instructional conference at Univ. of Sussex, Brighton, Sep. 1-17, 1965.
• Serge Lang, Algebraic number theory. GTM 110, Springer 1970, 2000

Revised on July 27, 2011 20:00:52 by Zoran Škoda (31.45.184.75)