alternative definition of valuation

One may use a slightly different version of the third criterion to define a slightly more general definition of valuationMathworldPlanetmathPlanetmath.

A valuation on a field 𝕂 is a map ||:𝕂 such that

  1. 1.

    |x|=0 if an only if x=0

  2. 2.


  3. 3.


The quantity C which appears in the third criterion is a positive real number which is known as the .

There is some flexibility in the choice of the constant C in this definition — one can replace C by a larger number C. To deal with this ambiguity, one defines the of the valuation as


The relationMathworldPlanetmath of this definition to the usual one is the following. On the one hand, if || satisfies the usual definition, then


so a valuation in the old sense is a valuation in the new sense with constant 2.

On the other hand, suppose that || satisfies the alternative definition with constant C<2. Then we have the following result.

TheoremMathworldPlanetmath If || is a valuation according to the definition of this entry with constant C2, then || satisfies the identity


The proof of this assertion is given in a supplement to this entry.

The foregoing discussion shows that the new definition is more general than the old definition precisely when C>2. However, this extra generalty is not as great as it might seem at first sight. As is obvious from examining the definition, if || is a valuation, then so is ||p for any power p>0. Furthermore, if the valuation has constant C, then valuation ||p has constant Cp. Therefore, given any valuation || in the sense of this entry, there will exist a number p such that ||p is a valuation in the sense of the parent entry. Moreover, given the fact that two valuations which are powers of each other are equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath, one sees that the extra generality is not that interesting since the new valuations are equivalent to the old valuations.

Title alternative definition of valuation
Canonical name AlternativeDefinitionOfValuation
Date of creation 2013-03-22 14:55:47
Last modified on 2013-03-22 14:55:47
Owner rspuzio (6075)
Last modified by rspuzio (6075)
Numerical id 7
Author rspuzio (6075)
Entry type Definition
Classification msc 11R99
Classification msc 12J20
Classification msc 13A18
Classification msc 13F30