Calculation of the Class Numbers of Imaginary Cyclic Quartic Fields

Abstract

Any imaginary cyclic quartic field can be expressed uniquely in the form $K = Q(\sqrt {A(D + B\sqrt D )} )$, where A is squarefree, odd and negative, $D = {B^2} + {C^2}$ is squarefree, $B > 0,C > 0$, and $(A,D) = 1$. 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 $h(K)$ of such fields K is described.