Below we will focus attention on number fields leaving the function field case to the reader.
The following lemma essentially says that the denominator of an element of a global field is only ``nontrivial'' at a finite number of valuations.
Suppose now that is a finite extension of , so satisfies a monic polynomial
Any valuation on a global field is either archimedean, or discrete non-archimedean with finite residue class field, since this is true of and and is a property preserved by extending a valuation to a finite extension of the base field. Hence it makes sense to talk of normalized valuations. Recall that the normalized -adic valuation on is , and if is a valuation on a number field equivalent to an extension of , then the normalization of is the composite of the sequence of maps
There are two extensions of to , since , as . The image of under each embedding into is a unit in , so the normalized valuation of is, in both cases, equal to . More generally, for any valuation of of characteristic an odd prime , the normalized valuation of is .
Since in two ways, there are exactly two normalized archimedean valuations on , and both of their values on equal . Notice that the product of the absolute values of with respect to all normalized valuations is
Let run through all normalized valuations of (or of ), and write if the restriction of to is equivalent to . Then by Theorem 19.2.2,
By multiplicativity of valuations, if the theorem is true for and then it is true for the product and quotient (when ). The theorem is clearly true for , which has valuation at all valuations. Thus to prove the theorem for it suffices to prove it when is a prime number. Then we have , , and for primes that . Thus
If is a valuation on a field , recall that we let denote the completion of with respect to . Also when is non-archimedean, let
We will use the following lemma later (see Lemma 20.3.3) to prove that formation of the adeles of a global field is compatible with base change.
Since for all , the left hand side of (20.1.2) is contained in the right hand side if for and . Thus by Lemma 20.1.2, for all but finitely many the left hand side of (20.1.2) is contained in the right hand side. We have just eliminated the finitely many primes corresponding to ``denominators'' of some , and now only consider such that for all .
For any elements , consider the discriminant
William Stein 2004-05-06