independence of p-adic valuations

We prove the following particular case:

Proposition 1.

Let p1,,pnZ be distinct prime numbersMathworldPlanetmath and let pi be the corresponding p-adic valuationsMathworldPlanetmathPlanetmath of Q. Let a1,,anZ and let ϵi be arbitrary positive real numbers, then there exists yZ such that for all i=1,,n:


Let p be an arbitrary prime, and let ϵ be an arbitrary positive real number. Notice that injects into p=lim/pn, the p-adic integers. For any b, we also write b for its image in p, and it can be written as a sequenceMathworldPlanetmathPlanetmath b=(bj) with bbjmodpj. Let n=np,ϵ be such that p-n<ϵ (and thus for any other c such that cbnmodpn we have b-cpp-n<ϵ).

Now, for the proof of the propositionPlanetmathPlanetmath, let ni=npi,ϵi and recall that by the Chinese Remainder TheoremMathworldPlanetmathPlanetmathPlanetmath we have an isomorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath:


Therefore we can find an element y~ of /(pini) (and thus a lift y of y~ to ) such that yaimodpini for all i=1,,n. Hence:


Title independence of p-adic valuations
Canonical name IndependenceOfPadicValuations
Date of creation 2013-03-22 14:12:14
Last modified on 2013-03-22 14:12:14
Owner alozano (2414)
Last modified by alozano (2414)
Numerical id 4
Author alozano (2414)
Entry type Corollary
Classification msc 11R99
Related topic Valuation
Related topic PAdicIntegers
Related topic PAdicValuation