Class Numbers of Real Cyclotomic Fields of Conductor pq

The class numbers h+ of the real cyclotomic fields are very hard to compute. Methods based on discriminant bounds become useless as the conductor of the field grows and that is why other methods have been developed, which approach the problem from different angles. In this thesis we extend a method of Schoof that was designed for real cyclotomic fields of prime conductor to real cyclotomic fields of conductor equal to the product of two distinct odd primes. Our method calculates the index of a specific group of cyclotomic units in the full group of units of the field. This index has h+ as a factor. We then remove from the index the extra factor that does not come from h+ and so we have the order of h+. We apply our method to real cyclotomic fields of conductor < 2000 and we test the divisibility of h+ by all primes < 10000. Finally, we calculate the full order of the l-part of h+ for all odd primes l < 10000.</italic

Ideal class groups of cyclotomic number fields II

We first study some families of maximal real subfields of cyclotomic fields with even class number, and then explore the implications of large plus class numbers of cyclotomic fields. We also discuss capitulation of the minus part and the behaviour of p-class groups in cyclic ramified p-extensions

Class numbers of totally real fields and applications to the Weber class number problem

The determination of the class number of totally real fields of large discriminant is known to be a difficult problem. The Minkowski bound is too large to be useful, and the root discriminant of the field can be too large to be treated by Odlyzko's discriminant bounds. We describe a new technique for determining the class number of such fields, allowing us to attack the class number problem for a large class of number fields not treatable by previously known methods. We give an application to Weber's class number problem, which is the conjecture that all real cyclotomic fields of power of 2 conductor have class number 1.Comment: Accepted for publication by Acta Arithmetic