On the class number of a real abelian field of prime conductor
Volume 199 / 2021
Acta Arithmetica 199 (2021), 145-152
MSC: Primary 11R29; Secondary 11R18.
DOI: 10.4064/aa191111-19-11
Published online: 24 May 2021
Abstract
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 $F$.
