Computation of the Ideal Class Group of Certain Complex Quartic Fields

The ideal class group of quartic fields $K = F(\sqrt \mu )$, where $F = {\mathbf {Q}}(i)$, is calculated by a method adapted from the method of cycles of reduced ideals for real quadratic fields. The class number is found in this way for 5000 fields $K = F(\sqrt \pi ),\pi \equiv \pm 1 \bmod 4,\pi$ a prime of F. A tabulation of the distribution of class numbers shows a striking similarity to that for real quadratic fields with prime discriminant. Also, two fields were found with noncyclic ideal class group $C(3) \times C(3)$.