Classification of Algebraic Function Fields with Divisor Class Number Two

In a previous paper we proved that there are 11 quadratic algebraic function fields with divisor class number two. Here we complete the classification of algebraic function fields with divisor class number two giving all non-quadratic solutions. Our result is the following. Let us denote bykthe finite field withqelements. Up to isomorphism, there are exactly 8 non-quadratic algebraic function fields of one variableK/khavingkfor full constant field and with a divisor class number equal to two.