Divisibility of class numbers of certain families of quadratic fields

TL;DR

This paper constructs specific families of quadratic fields with class numbers divisible by 3, using a trinomial and parametrization techniques, and verifies the divisibility through explicit computations.

Contribution

It introduces new families of quadratic fields with class number divisibility by 3 using novel algebraic tools and parametrizations.

Findings

Constructed families of quadratic fields with class number divisible by 3.

Verified divisibility through explicit class number computations.

Demonstrated the effectiveness of Kishi and Miyake's parametrization methods.

Abstract

We construct some families of quadratic fields whose class numbers are divisible by $3.$ The main tools used are a trinomial introduced by Kishi and a parametrization of Kishi and Miyake of a family of quadratic fields whose class numbers are divisible by $3.$ At the end we compute class number of these fields for some small values and verify our results.