proof of Ostrowski’s valuation theorem
This article proves Ostrowski’s theorem on valuations of , which states:
We start with an estimation lemma:
If are integers and any nontrivial absolute value on , then .
Proof. Write for , and with . Then clearly
by the triangle inequality; also, .
Now let ; the first two factors each approach , and the lemma follows.
Proof of Ostrowski’s theorem:
First assume that for every we have . Then by the lemma, , so that for every we have
Since this holds for every , after reversing the roles of , we see that in fact equality holds, so that for every , and for some constant ; this absolute value is obviously equivalent to .
If instead, for some we have , then by the lemma, for every , . Thus the absolute value is nonarchimedean. Define and let be the (unique) maximal ideal defined by . Then since for every , and is nonzero since otherwise the valuation would be trivial (we would have for every ). Thus is prime since is, so is equal to for some rational prime . Now, if for an integer , then cannot be strictly less than (else it would be in ), so and . But given any , we can write with prime to , so that
so that the valuation is obviously equivalent to the -adic valuation.
|Title||proof of Ostrowski’s valuation theorem|
|Date of creation||2013-03-22 17:58:26|
|Last modified on||2013-03-22 17:58:26|
|Last modified by||rm50 (10146)|