# 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 19.2.1   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 class group of the number field is by definition the quotient of by the image of .

Theorem 19.2.2   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 19.1.12), hence compact. But a compact discrete group is finite. Subsections
William Stein 2012-09-24