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On divisibility of the class number $h^+$ of the real cyclotomic fields of prime degree $l$


Author: Stanislav Jakubec
Journal: Math. Comp. 67 (1998), 369-398
MSC (1991): Primary 11R29
DOI: https://doi.org/10.1090/S0025-5718-98-00916-8
MathSciNet review: 1443121
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Abstract: In this paper, criteria of divisibility of the class number $h^+$ of the real cyclotomic field $\mathbf {Q}(\zeta _p+\zeta _p^{-1})$ of a prime conductor $p$ and of a prime degree $l$ by primes $q$ the order modulo $l$ of which is $\frac {l-1}{2}$, are given. A corollary of these criteria is the possibility to make a computational proof that a given $q$ does not divide $h^+$ for any $p$ (conductor) such that both $\frac {p-1}{2},\frac {p-3}{4}$ are primes. Note that on the basis of Schinzel’s hypothesis there are infinitely many such primes $p$.


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Stanislav Jakubec
Affiliation: Mathematical Institute of the Slovak Academy of Sciences, Štefánikova 49, 814 73 Bratislava, Slovakia
Email: jakubec@mau.savba.sk

Received by editor(s): March 16, 1995
Received by editor(s) in revised form: April 12, 1996
Article copyright: © Copyright 1998 American Mathematical Society