p-adic valuation

Let p be a positive prime numberMathworldPlanetmath.  For every non-zero rational numberPlanetmathPlanetmathPlanetmath x there exists a unique integer n such that


with some integers u and v indivisible by p.  We define


obtaining a non-trivial (http://planetmath.org/TrivialValuation) non-archimedean valuation, the so-called p-adic valuationMathworldPlanetmathPlanetmath


of the field .

The value group of the p-adic valuation consists of all integer-powers of the prime number p.  The valuation ringMathworldPlanetmathPlanetmath of the valuation is called the ring of the p-integral rational numbers; their denominators, when reduced (http://planetmath.org/Fraction) to lowest terms, are not divisible by p.

The field of rationals has the 2-adic, 3-adic, 5-adic, 7-adic and so on valuations (which may be called, according to Greek, dyadic, triadic, pentadic, heptadic and so on).  They all are non-equivalent (http://planetmath.org/EquivalentValuations) with each other.

If one replaces the number 1p by any positive ϱ less than 1, one obtains an equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath (http://planetmath.org/EquivalentValuations) p-adic valuation; among these the valuation with  ϱ=1p  is sometimes called the normed p-adic valuation.  Analogously we can say that the absolute valueMathworldPlanetmathPlanetmath is the normed archimedean valuation of which corresponds the infinite prime of .

The productPlanetmathPlanetmath of all normed valuations of is the trivial valuation||tr,  i.e.

Title p-adic valuation
Canonical name PadicValuation
Date of creation 2013-03-22 14:55:50
Last modified on 2013-03-22 14:55:50
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 14
Author pahio (2872)
Entry type Definition
Classification msc 13A18
Synonym p-adic valuation
Related topic IndependenceOfPAdicValuations
Related topic IntegralElement
Related topic OrderValuation
Related topic StrictDivisibility
Defines p-integral rational number
Defines normed p-adic valuation
Defines normed archimedean valuation
Defines dyadic valuation
Defines triadic valuation
Defines pentadic valuation
Defines heptadic valuation