extension of valuation from complete base field

Here the valuationsMathworldPlanetmathPlanetmath are of rank one, and it may be supposed that the values are real numbers.

  • Assume a finite field extension K/k and a valuation of K.  If the base fieldMathworldPlanetmath is completePlanetmathPlanetmathPlanetmathPlanetmath (http://planetmath.org/Complete) with regard to this valuation, so is also the extension field.

  • If K/k is an algebraic field extension and if the base field k is complete (http://planetmath.org/Complete) with regard to its valuation  ||,   then this valuation has one and only one extensionPlanetmathPlanetmath to the field K.  This extension is determined by


    where N(α) is the norm of the element α in the simple field extension k(α)/k and n is the degree of this field extension.

These theorems concern also Archimedean valuations.

Title extension of valuation from complete base field
Canonical name ExtensionOfValuationFromCompleteBaseField
Date of creation 2013-03-22 15:01:01
Last modified on 2013-03-22 15:01:01
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 9
Author pahio (2872)
Entry type Theorem
Classification msc 13F30
Classification msc 13A18
Classification msc 12J20
Classification msc 11R99
Related topic CompleteUltrametricField
Related topic ValueGroupOfCompletion
Related topic NthRoot