A family of pairs of imaginary cyclic fields of degree $(p-1)/2$ with both class numbers divisible by $p$

Abstract

We construct a new infinite family of pairs of imaginary cyclic fields of degree $(p-1)/2$ explicitly with both class numbers divisible by a given prime number $p$. For the proof, we use the fundamental unit of $\mathbb{Q}(\sqrt{p})$, certain units which are roots of a parametric quartic polynomial, the Kummer theory, the Gauss sums and the Jacobi sums, linear recurrence sequences, a consequence of the Weil conjecture and a result of Lenstra which is a generalization of Artin conjecture on primitive roots. Our result is based on the famous Scholz' results on pairs of quadratic fields $\mathbb{Q}(\sqrt{D})$ and $\mathbb{Q}(\sqrt{-3D})$.