  We analyse the complexity of solving the discrete logarithm problem and of
testing the principality of ideals in a certain class of number fields. We
achieve the subexponential complexity in $O(L(1/3,O(1)))$ when both the
discriminant and the degree of the extension tend to infinity by using
techniques due to Enge, Gaudry and Thom\'{e} in the context of algebraic curves
over finite fields.
