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Mathematics of Computation

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Class numbers of real cyclotomic fields of prime conductor

Author: René Schoof
Journal: Math. Comp. 72 (2003), 913-937
MSC (2000): Primary 11R18, 11Y40
Published electronically: February 15, 2002
MathSciNet review: 1954975
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Abstract: The class numbers $h_{l}^{+}$ of the real cyclotomic fields $\mathbf {Q}(\zeta _{l}^{}+\zeta _{l}^{-1})$ are notoriously hard to compute. Indeed, the number $h_{l}^{+}$ is not known for a single prime $l\ge 71$. In this paper we present a table of the orders of certain subgroups of the class groups of the real cyclotomic fields $\mathbf {Q}(\zeta _{l}^{}+\zeta _{l}^{-1})$ for the primes $l<10,000$. It is quite likely that these subgroups are in fact equal to the class groups themselves, but there is at present no hope of proving this rigorously. In the last section of the paper we argue —on the basis of the Cohen-Lenstra heuristics— that the probability that our table is actually a table of class numbers $h_{l}^{+}$, is at least $98\%$.

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René Schoof
Affiliation: Dipartimento di Matematica, $2^{\mathrm {a}}$ Università di Roma “Tor Vergata", I-00133 Roma, Italy

Received by editor(s): November 7, 2000
Received by editor(s) in revised form: July 9, 2001
Published electronically: February 15, 2002
Article copyright: © Copyright 2002 American Mathematical Society