Calculation of the class numbers of imaginary cyclic quartic fields

Abstract

Any imaginary cyclic quartic field can be expressed uniquely in the form K = Q ( A ( D + B D ) ) K = Q(\sqrt {A(D + B\sqrt D )} ) , where A is squarefree, odd and negative, D = B 2 + C 2 D = {B^2} + {C^2} is squarefree, B > 0 , C > 0 B > 0,C > 0 , and ( A , D ) = 1 (A,D) = 1 . Explicit formulae for the discriminant and conductor of K are given in terms of A, B, C, D. The calculation of tables of the class numbers h ( K ) h(K) of such fields K is described.