The following theorem asserts that inequivalent valuations are in fact almost totally independent. For our purposes it will be superseded by the strong approximation theorem (Theorem 18.4.4).

If and the are -adic valuations, Theorem 14.3.1 is related to the Chinese Remainder Theorem (Theorem 5.1.4), but the strong approximation theorem (Theorem 18.4.4) is the real generalization.

and for

where
. For then as
, we have

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 similarly a such that

and

Then
will do.

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

and for

By the case for there is a such that

and

Then put

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 2012-09-24