discrete valuation

A discrete valuationPlanetmathPlanetmath on a field K is a valuationMathworldPlanetmath ||:K whose image is a discrete subset of .

For any field K with a discrete valuation ||, the set


is a subring of K with sole maximal idealMathworldPlanetmath


and hence R is a discrete valuation ring. Conversely, given any discrete valuation ring R, the field of fractionsMathworldPlanetmath K of R admits a discrete valuation sending each element xR to cn, where 0<c<1 is some arbitrary fixed constant and n is the order of x, and extending multiplicatively to K.

Note: Discrete valuations are often written additively instead of multiplicatively; under this alternate viewpoint, the element x maps to logc|x| (in the above notation) instead of just |x|. This transformation reverses the order of the absolute valuesMathworldPlanetmathPlanetmath (since c<1), and sends the element 0K to . It has the advantage that every valuation can be normalized by a suitable scalar multiple to take values in the integers.

Title discrete valuation
Canonical name DiscreteValuation
Date of creation 2013-03-22 13:59:14
Last modified on 2013-03-22 13:59:14
Owner djao (24)
Last modified by djao (24)
Numerical id 6
Author djao (24)
Entry type Definition
Classification msc 13F30
Classification msc 12J20
Synonym rank one valuations
Synonym rank-one valuations
Related topic DiscreteValuationRing
Related topic Valuation