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 ).
Assume first that all . By the Chinese Remainder Theorem,
Thus the map
In the general case, assume wlog that we are given a set of primes of and integers , and a set of primes with integers . First choose (using the case already proved above) so that
Now, there are only a finite number of primes such that is not the same as any of the and . Let . Again using the case proved above, choose such that
Then is the required element. ∎
|Title||weak approximation theorem|
|Date of creation||2013-03-22 18:35:21|
|Last modified on||2013-03-22 18:35:21|
|Last modified by||rm50 (10146)|