Cyclotomic Integers of Prescribed Absolute Value and the Class Group Original Research Article

We obtain a new method for the study of class groups of cyclotomic fields by investigating cyclotomic integers of prescribed absolute value. Explicit subgroups of the classgroupCmodulo the class groupC+of the maximal real subfield are exhibited and lower bounds on their orders are derived. For themth cyclotomic fieldKm, wherem=pam′, (p, m′)=1, andpis a prime, we determine the structure ofC+CP/C+CQup to a binary parameter; hereCP,CQare the subgroups ofCgenerated by the classes [Pi] respectively [Qi], wherepfactors inKmas ∏ Qi,Qi=Pphi(pa)i, and thePiare prime ideals.