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Class numbers of real cyclotomic fields of composite conductor

Part of: Computational number theory Algebraic number theory: global fields Diophantine approximation, transcendental number theory

Published online by Cambridge University Press:  01 August 2014

John C. Miller
John C. Miller*
Affiliation:
Department of Mathematics, Rutgers University, Hill Center for the Mathematical Sciences, 110 Frelinghuysen Road, Piscataway, NJ 08854-8019, USA email [email protected]

Abstract

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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.

MSC classification

Secondary: 11R18: Cyclotomic extensions 11R29: Class numbers, class groups, discriminants 11J71: Distribution modulo one 11R80: Totally real fields 11Y40: Algebraic number theory computations
Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 17 , Special Issue A: Algorithmic Number Theory Symposium XI , 2014 , pp. 404 - 417
Copyright
© The Author 2014 

References

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