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

2018/09/21 by Miho Aoki, Aoki, Miho, Yasuhiro Kishi +1
Computer Science · Mathematics · #11R11 #11R16 #11R29 #Algebraic Geometry and Number Theory #FOS: Mathematics #Number Theory (math.NT) #Topological and Geometric Data Analysis

paper · pdf · doi:10.48550/arxiv.1809.07982

openalex publication_date 2018/09/21 · openalex created_date 2019/06/27 · openalex updated_date 2026/07/28

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(√(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(√(D)) and \mathbb Q (√(-3D)).

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