Monogenic trinomials and class numbers of related quadratic fields

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 $\{1,\theta ,{\theta }^2,\dots ,{\theta }^{N-1}\}$ 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 $f(x)$.