Ideals and Divisors

Suppose that is a finite extension of . Let be the the free abelian group on a set of symbols in bijection with the non-archimedean valuation  of . Thus an element of is a formal linear combination

where and all but finitely many are 0.

Lemma 21.2.1   There is a natural bijection between and the group of nonzero fractional ideals of . The correspondence is induced by

where is a non-archimedean valuation.

Endow with the discrete topology. Then there is a natural continuous map given by

This map is continuous since the inverse image of a valuation (a point) is the product

which is an open set in the restricted product topology on . Moreover, the image of in is the group of nonzero principal fractional ideals.

Recall that the of the number field is by definition the quotient of by the image of .

Theorem 21.2.2   The class group of a number field is finite.

Proof. We first prove that the map is surjective. Let be an archimedean valuation on . If is a non-archimedean valuation, let be a -idele such that at ever valuation  except  and . At , choose to be a generator for the maximal ideal of , and choose to be such that . Then and , so . Also maps to .

Thus the group of ideal classes is the continuous image of the compact group (see Theorem 21.1.12), hence compact. But a compact discrete group is finite.

Subsections
William Stein 2004-05-06