1,132 research outputs found
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
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
Numerical verification of the Cohen-Lenstra-Martinet heuristics and of Greenberg's -rationality conjecture
In this paper we make a series of numerical experiments to support
Greenberg's -rationality conjecture, we present a family of -rational
biquadratic fields and we find new examples of -rational multiquadratic
fields. In the case of multiquadratic and multicubic fields we show that the
conjecture is a consequence of the Cohen-Lenstra-Martinet heuristic and of the
conjecture of Hofmann and Zhang on the -adic regulator, and we bring new
numerical data to support the extensions of these conjectures. We compare the
known algorithmic tools and propose some improvements
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