For those familiar with algebraic geometry and algebraic curves, one
can prove Theorem 19.2.3 from an alternative point of
view. There is a bijection between nonsingular geometrically
irreducible projective curves over
and function fields over
(which we assume are finite separable extensions of
such
that
). Let be the curve corresponding to .
The group is in bijection with the divisors of degree 0 on
, a group typically denoted
. The quotient of
by principal divisors is denoted
. The * Jacobian* of is an abelian variety
over the finite
field
whose dimension is equal to the genus of . Moreover,
assuming has an
-rational point, the elements of
are in natural bijection with the
-rational points on . In
particular, with these hypothesis, the class group of , which is
isomorphic to
, is in bijection with the group of
-rational points on an algebraic variety over a finite field.
This gives an alternative more complicated proof of finiteness of the
degree 0 class group of a function field.
Without the degree 0 condition, the divisor class group won't be finite. It
is an extension of
by a finite group.

where is the greatest common divisor of the degrees of
elements of , which is when has a rational
point.

William Stein
2012-09-24