Real and imaginary quadratic representations of hyperelliptic function fields

A hyperelliptic function field can be always be represented as a real quadr atic extension of the rational function field. If at least one of the ratio nal prime divisors is rational over the field of constants, then it also ca n be represented as an imaginary quadratic extension of the rational functi on field. The arithmetic in the divisor class group can be realized in the second case by Canter's algorithm. We show that in the first case one can c ompute in the divisor class group of the function field using reduced ideal s and distances of ideals in the orders involved. Furthermore, we show how the two representations are connected and compare the computational complex ity.