## The Function Field Case

When is a finite separable extension of , we define the divisor group of to be the free abelian group on all the valuations . For each the number of elements of the residue class field of is a power, say , of the number  of elements in . We call the degree of , and similarly define to be the degree of the divisor . The divisors of degree 0 form a group . As before, the principal divisor attached to is . The following theorem is proved in the same way as Theorem 19.2.2.

Theorem 19.2.3   The quotient of modulo the principal divisors is a finite group.

William Stein 2012-09-24