weak approximation theorem

The weak approximation theorem allows selection, in a Dedekind ring, of an element having specific valuationsMathworldPlanetmathPlanetmath at a specific finite setMathworldPlanetmath of primes, and nonnegative valuations at all other primes. It is essentially a generalizationPlanetmathPlanetmath of the Chinese Remainder theoremMathworldPlanetmathPlanetmathPlanetmath, as is evident from its proof.

Theorem 1 (Weak ).

Let A be a Dedekind domainMathworldPlanetmath with fraction field K. Then for any finite set p1,,pk of primes of A and integers a1,,ak, there is xK such that νpi((x))=ai and for all other prime idealsMathworldPlanetmathPlanetmathPlanetmath p, νp((x))0. Here νp is the p-adic valuation associated with a prime ideal p.


Assume first that all ai0. By the Chinese Remainder Theorem,


Thus the map


is surjectivePlanetmathPlanetmath. Now choose xipiai,xipiai+1; this is possible since these two ideals are unequal by unique factorizationMathworldPlanetmath. Choose xA with image (x1,,xk). Clearly ν𝔭i((x))=ai. But xA, so all other valuations are nonnegative.

In the general case, assume wlog that we are given a set 𝔭1,,𝔭r of primes of A and integers a1,,ar0, and a set 𝔮1,,𝔮t of primes with integers b1,,bt<0. First choose yK (using the case already proved above) so that


Now, there are only a finite number of primes 𝔭k such that 𝔭k is not the same as any of the 𝔮j and ν𝔭k((y))>0. Let ν𝔭k((y))=ck>0. Again using the case proved above, choose xK such that


Then x/y is the required element. ∎

Title weak approximation theorem
Canonical name WeakApproximationTheorem
Date of creation 2013-03-22 18:35:21
Last modified on 2013-03-22 18:35:21
Owner rm50 (10146)
Last modified by rm50 (10146)
Numerical id 6
Author rm50 (10146)
Entry type Theorem
Classification msc 13F05
Classification msc 11R04
Related topic IndependenceOfTheValuations
Related topic ChineseRemainderTheoremInTermsOfDivisorTheory