1,194 research outputs found
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
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