Imaginary cyclic fields of degree $p - 1$ whose relative class numbers are divisible by $p$

Abstract

We give a sufficient condition for an imaginary cyclic field of degree $p - 1$ containing $\mathbf{Q}(\zeta + \zeta^{-1})$ to have the relative class number divisible by $p$. As a consequence, we see that there exist infinitely many imaginary cyclic fields of degree $p - 1$ with the relative class number divisible by $p$.