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On Divisibility of the Class Number of a Real Abelian Field and Fermat Quotients | Synapse

May 24, 2026Open Access

On Divisibility of the Class Number of a Real Abelian Field and Fermat Quotients

HIHumio IchimuraIbaraki University

Key Points

This research aims to establish improved conditions for the divisibility of class numbers in real abelian fields based on Fermat quotients.
Building upon previous work by Jakubec and Metsnkyl
Employing different mathematical techniques to refine existing conditions
Defines new sufficient conditions for r h K, where r is an odd prime and h K is the class number of K.
Clarifies relationships between the class number and Fermat quotients associated with cyclotomic elements.

Abstract

Let r be an odd prime number, and let K be a real abelian field of conductor f K and class number h K .When r f K , Jakubec (Abh.Math.Semin.Univ.Hambg.63 (1993), 67-86) and Metsnkyl (Manuscripta Math.93 (1997), 481-498) gave a sufficient condition for r h K in terms related to Fermat quotients associated to cyclotomic elements m -1 for integers m dividing f K .We give a refined version of this result by different methods.

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Cite This Study

Humio Ichimura (Wed,) studied this question.

synapsesocial.com/papers/6a12953848a0ea16656715b8 https://doi.org/https://doi.org/10.2206/kyushujm.79.257