trivial valuation

The trivial valuation of a field K is the Krull valuation||  of K such that  |0|=0  and  |x|=1  for other elements x of K.


  1. 1.

    Every field has the trivial valuation.

  2. 2.

    The trivial valuation is non-archimedean.

  3. 3.

    The valuation ringMathworldPlanetmathPlanetmath of the trivial valuation is the whole field and the corresponding maximal idealPlanetmathPlanetmath is the zero idealPlanetmathPlanetmath.

  4. 4.

    The field is completePlanetmathPlanetmathPlanetmathPlanetmath ( with respect to (the metric given by) its trivial valuation.

  5. 5.

    A finite fieldMathworldPlanetmath has only the trivial valuation.  (Let a be the primitive elementMathworldPlanetmathPlanetmath of the multiplicative groupMathworldPlanetmath of the field, which is cyclic (  If  ||  is any valuation of the field, then one must have  |a|=1  since otherwise  |1|1.  Consequently,  |x|=|am|=|a|m=1m=1  for all non-zero elements x.)

  6. 6.

    Every algebraic extensionMathworldPlanetmath of finite fields has only the trivial valuation, but every field of characteristicPlanetmathPlanetmath 0 has non-trivial valuations.

Title trivial valuation
Canonical name TrivialValuation
Date of creation 2013-03-22 14:20:23
Last modified on 2013-03-22 14:20:23
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 16
Author pahio (2872)
Entry type Definition
Classification msc 12J20
Classification msc 11R99
Related topic IndependenceOfTheValuations
Related topic KrullValuation