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On the class number of a real abelian field of prime conductor

Volume 199 / 2021

Humio Ichimura Acta Arithmetica 199 (2021), 145-152 MSC: Primary 11R29; Secondary 11R18. DOI: 10.4064/aa191111-19-11 Published online: 24 May 2021


For a fixed integer $n \geq 1$, let $p=2n\ell +1$ be a prime number with an odd prime number $\ell $ and let $F=F_{p,\ell }$ be the real abelian field of conductor $p$ and degree $\ell $. We prove that for each fixed $n$, there exist only finitely many pairs $(\ell ,r)$ of prime numbers $\ell $ and $r$ such that (a) $p=2n\ell +1$ is a prime number, (b) $r$ is a primitive root modulo $\ell $ and (c) $r$ divides the class number $h_F$ of $F$.


  • Humio IchimuraFaculty of Science, Ibaraki University
    Bunkyo 2-1-1, Mito, 310-8512, Japan

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