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
and all but finitely many are 0.
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
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
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 .
We first prove that the map
is surjective. Let
be an archimedean valuation on
non-archimedean valuation, let
at ever valuation
to be a generator for the maximal ideal of
, and choose
to be such that
Thus the group of ideal classes is the continuous image of the
(see Theorem 21.1.12), hence
compact. But a compact discrete group is finite.