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.

William Stein 2004-05-06