nLab
discrete valuation

There are two different notions of a discrete valuation.

A discrete valuation on a field KK is a function v:KZ{}v:K\to \mathbf{Z}\cup \{\infty\} such that

  • vv defines the homomorphism of groups v| K:K *Zv|_K : K^*\to \mathbf{Z} where K *K^* is the multiplicative group of KK

  • v(0)=v(0) = \infty

  • v(x+y)inf{v(x),v(y)}v(x+y) \geq inf\{ v(x),v(y)\}

with usual conventions for \infty.

The set R v={xK|v(x)0}R_v = \{x\in K\,|\, v(x)\geq 0\} is called the valuation ring of the valuation vv, and it is an integral domain with quotient field KK; its part 𝔭 v={xK|v(x)>0}\mathfrak{p}_v = \{x\in K\,|\, v(x)\gt 0\} is a maximal ideal of R vR_v and is called the valuation ideal of vv. This can be generalized to other valuations.

Let 0<ρ<10\lt \rho\lt 1 and define |x| v=ρ v(x)|x|_v = \rho^{v(x)}. Then |||| will be a nonarchimedean multiplicative discrete valuation in the sense of the following definition. If we change a ρ\rho then we get an equivalent multiplicative valuation.

A valuation on a field KK is a function ||:KR 0| |:K\to \mathbf{R}_{\geq 0} such that

  • |x|=0|x| = 0 iff x=0x=0

  • |xy|=|x||y||x y| = |x| |y|

  • there is a constant CC such that |1+x|C|1+x|\leq C if |x|1|x|\leq 1

Two valuations on the same field are equivalent if one is the positive power of another i.e. c>0\exists c\gt 0 such that |x| 1=|x| 2 c|x|_1 = |x|_2^c for all xKx\in K.

A valuation is non-archimedean if CC above can be taken 11 and archimedean otherwise. A (multiplicative) valuation is discrete if there is a neighborhood U1R +U\ni 1\in \mathbf{R}_+ such that the only xKx\in K such that |x|U|x|\in U is 1 KK1_K\in K.

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