# Weak Approximation

The following theorem asserts that inequivalent valuations are in fact almost totally indepedent. For our purposes it will be superseded by the strong approximation theorem (Theorem 20.4.4).

Theorem 16.3.1 (Weak Approximation)   Let , for , be inequivalent nontrivial valuations of a field . For each , let be the topological space consisting of the set of elements of with the topology induced by . Let be the image of in the topological product

equipped with the product topology. Then is dense in .

The conclusion of the theorem may be expressed in a less topological manner as follows: given any , for , and real , there is an such that simultaneously

If and the are -adic valuations, Theorem 16.3.1 is related to the Chinese Remainder Theorem (Theorem 9.1.3), but the strong approximation theorem (Theorem 20.4.4) is the real generalization.

Proof. We note first that it will be enough to find, for each , an element such that

and     for

where . For then as , we have

It is then enough to take

By symmetry it is enough to show the existence of with

and   for

We will do this by induction on .

First suppose . Since and are inequivalent (and all absolute values are assumed nontrivial) there is an such that

 and (16.2)

and similarly a such that

and

Then will do.

Remark 16.3.2   It is not completely clear that one can choose an  such that (16.3.1) is satisfied. Suppose it were impossible. Then because the valuations are nontrivial, we would have that for any if then . This implies the converse statement: if and then . To see this, suppose there is an such that and . Choose such that . Then for any integer we have , so by hypothesis . Thus for all . Since we have as , so , a contradiction since . Thus if and only if , and we have proved before that this implies that is equivalent to .

Next suppose . By the case , there is an such that

and   for

By the case for there is a such that

and

Then put

where is sufficiently large so that and for .

Example 16.3.3   Suppose , let be the archimedean absolute value and let be the -adic absolute value. Let , , and , as in the remark right after Theorem 16.3.1. Then the theorem implies that there is an element such that

and

As in the proof of the theorem, we can find such a by finding a such that and , and a and . For example, and works, since and and and . Again following the proof, we see that for sufficiently large we can take

We have , , , , , . None of the work for , but works.

William Stein 2004-05-06