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单词 ExtensionOfValuationFromCompleteBaseField
释义

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

    |α|=|N(α)|n(αK),

    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.

Titleextension of valuation from complete base field
Canonical nameExtensionOfValuationFromCompleteBaseField
Date of creation2013-03-22 15:01:01
Last modified on2013-03-22 15:01:01
Ownerpahio (2872)
Last modified bypahio (2872)
Numerical id9
Authorpahio (2872)
Entry typeTheorem
Classificationmsc 13F30
Classificationmsc 13A18
Classificationmsc 12J20
Classificationmsc 11R99
Related topicCompleteUltrametricField
Related topicValueGroupOfCompletion
Related topicNthRoot
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更新时间:2025/5/4 11:40:39