discrete valuation

There are two different notions of a discrete valuation.

A discrete valuation on a field K is a function v:KZ{} such that

  • v defines the homomorphism of groups v K:K *Z where K * is the multiplicative group of K

  • v(0)=

  • v(x+y)inf{v(x),v(y)}

with usual conventions for .

The set R v={xKv(x)0} is called the valuation ring of the valuation v, and it is an integral domain with quotient field K; its part 𝔭 v={xKv(x)>0} is a maximal ideal of R v and is called the valuation ideal of v. This can be generalized to other valuations.

Let 0<ρ<1 and define x v=ρ v(x). Then will be a nonarchimedean multiplicative discrete valuation in the sense of the following definition. If we change a ρ then we get an equivalent multiplicative valuation.

A valuation on a field K is a function :KR 0 such that

  • x=0 iff x=0

  • xy=xy

  • there is a constant C such that 1+xC if x1

Two valuations on the same field are equivalent if one is the positive power of another i.e. c>0 such that x 1=x 2 c for all xK.

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 U1R + such that the only xK such that xU is 1 KK.

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 (