On the classification of algebraic function fields of class number three

Let F be an algebraic function field of one variable having a finite field F q with q > 2 elements as its field of constants. We determine all such fields for which the class number is three. More precisely, we show that, up to F q -isomorphism, there are only 8 of such function fields. For q = 2 the problem has been solved under the additional hypothesis that the function field is quadratic.