Class numbers of real cyclotomic fields of composite conductor

Until recently, the ‘plus part’ of the class numbers of cyclotomic fields had only been determined for fields of root discriminant small enough to be treated by Odlyzko’s discriminant bounds. However, by finding lower bounds for sums over prime ideals of the Hilbert class field, we can now establish upper bounds for class numbers of fields of larger discriminant. This new analytic upper bound, together with algebraic arguments concerning the divisibility properties of class numbers, allows us to unconditionally determine the class numbers of many cyclotomic fields that had previously been untreatable by any known method. In this paper, we study in particular the cyclotomic fields of composite conductor.