The Function Field Case

When $K$ is a finite separable extension of $\mathbf{F}(t)$, we define the divisor group $D_K$ of $K$ to be the free abelian group on all the valuations $v$. For each $v$ the number of elements of the residue class field $\mathbf{F}_v = \O _v/\wp_v$ of $v$ is a power, say $q^{n_v}$, of the number $q$ of elements in $\mathbf{F}_v$. We call $n_v$ the degree of $v$, and similarly define $\sum n_v d_v$ to be the degree of the divisor $\sum n_v\cdot v$. The divisors of degree 0 form a group $D_K^0$. As before, the principal divisor attached to $a\in K^*$ is $\sum \ord _v(a) \cdot v \in D_K$. The following theorem is proved in the same way as Theorem 21.2.2.

Theorem 21.2.3   The quotient of $D_K^0$ modulo the principal divisors is a finite group.