Monogenic trinomials and class numbers of related quadratic fields

TL;DR

This paper explores the relationship between monogenic trinomials and the divisibility properties of class numbers in related quadratic fields, providing new insights into algebraic number theory and the structure of rings of integers.

Contribution

It investigates how certain monogenic trinomials influence the divisibility of class numbers of associated quadratic fields, a novel connection in algebraic number theory.

Findings

Identifies conditions under which class numbers are divisible by specific integers.

Establishes links between polynomial monogenicity and quadratic field class number divisibility.

Provides examples of monogenic trinomials affecting class number properties.

Abstract

We say that a monic polynomial $f(x)\in {\mathbb Z}[x]$ of degree $N\ge 2$ is monogenic if $f(x)$ is irreducible over ${\mathbb Q}$ and $\{1,\theta,\theta^2,\ldots ,\theta^{N-1}\}$ is a basis for the ring of integers of ${\mathbb Q}(\theta)$, where $f(\theta)=0$. In this article, we investigate the divisibility of the class numbers of quadratic fields ${\mathbb Q}(\sqrt{\delta})$ for certain families of monogenic trinomials $f(x)=x^N+Ax+B$, where $\delta\ne \pm 1$ is a squarefree divisor of the discriminant of $f(x)$.