weak approximation theorem

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

Theorem 1 (Weak ).

Let $A$ be a Dedekind domain with fraction field $K$ . Then for any finite set $\mathfrak{p}_{1},\ldots,\mathfrak{p}_{k}$ of primes of $A$ and integers $a_{1},\ldots,a_{k}$ , there is $x\in K^{\star}$ such that $\nu_{\mathfrak{p}_{i}}((x))=a_{i}$ and for all other prime ideals $\mathfrak{p}$ , $\nu_{\mathfrak{p}}((x))\geq 0$ . Here $\nu_{\mathfrak{p}}$ is the $\mathfrak{p}$ -adic valuation associated with a prime ideal $\mathfrak{p}$ .

Proof.

Assume first that all $a_{i}\geq 0$ . By the Chinese Remainder Theorem,

A/\mathfrak{p}_{1}^{a_{1}+1}\times\cdots A/\mathfrak{p}_{k}^{a_{k}+1}\cong A/% \mathfrak{p}_{1}^{a_{1}+1}\cdots\mathfrak{p}_{k}^{a_{k}+1}

Thus the map

A\to A/\mathfrak{p}_{1}^{a_{1}+1}\times\cdots A/\mathfrak{p}_{k}^{a_{k}+1}

is surjective. Now choose $x_{i}\in p_{i}^{a_{i}},x_{i}\notin p_{i}^{a_{i}+1}$ ; this is possible since these two ideals are unequal by unique factorization. Choose $x\in A$ with image $(x_{1},\ldots,x_{k})$ . Clearly $\nu_{\mathfrak{p}_{i}}((x))=a_{i}$ . But $x\in A$ , so all other valuations are nonnegative.

In the general case, assume wlog that we are given a set $\mathfrak{p}_{1},\ldots,\mathfrak{p}_{r}$ of primes of $A$ and integers $a_{1},\ldots,a_{r}\geq 0$ , and a set $\mathfrak{q}_{1},\ldots,\mathfrak{q}_{t}$ of primes with integers $b_{1},\ldots,b_{t}<0$ . First choose $y\in K^{\star}$ (using the case already proved above) so that

\begin{cases}\nu_{\mathfrak{p}}((y))=0&\mathfrak{p}=\mathfrak{p}_{i}\\ \nu_{\mathfrak{p}}((y))=-b_{i}&\mathfrak{p}=\mathfrak{q}_{j}\\ \nu_{\mathfrak{p}}((y))\geq 0&\text{otherwise}\end{cases}

Now, there are only a finite number of primes $\mathfrak{p}^{\prime}_{k}$ such that $\mathfrak{p}^{\prime}_{k}$ is not the same as any of the $\mathfrak{q}_{j}$ and $\nu_{\mathfrak{p}^{\prime}_{k}}((y))>0$ . Let $\nu_{\mathfrak{p}^{\prime}_{k}}((y))=c_{k}>0$ . Again using the case proved above, choose $x\in K^{\star}$ such that

\begin{cases}\nu_{\mathfrak{p}}((x))=a_{i}&\mathfrak{p}=\mathfrak{p}_{i}\\ \nu_{\mathfrak{p}}((x))=0&\mathfrak{p}=\mathfrak{q}_{j}\\ \nu_{\mathfrak{p}}((x))=c_{k}&\mathfrak{p}=\mathfrak{p}^{\prime}_{k}\\ \nu_{\mathfrak{p}}((x))\geq 0&\text{otherwise}\end{cases}

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