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

TL;DR

This paper constructs an infinite family of pairs of imaginary cyclic fields of degree (p-1)/2 with both class numbers divisible by a prime p, using advanced number theory techniques and generalizing classical results.

Contribution

It introduces a new explicit construction of such pairs of fields with class number divisibility properties, extending Scholz's classical results.

Findings

Constructed infinite families of pairs of imaginary cyclic fields with class numbers divisible by p

Utilized units, Kummer theory, and sums to establish divisibility properties

Generalized classical results on quadratic fields to higher degree fields

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 $\mathbb Q (\sqrt{-3D})$.