Monogenic $S_3$-cubic polynomials and class numbers of associated quadratic fields

Lenny Jones

Published 2024-12-29Version 1

We say that a monic polynomial $f(x)\in {\mathbb Z}[x]$ is monogenic if $f(x)$ is irreducible over ${\mathbb Q}$ and $\{1,\theta,\theta^2,\ldots ,\theta^{\deg{f}-1}\}$ is a basis for the ring of integers of ${\mathbb Q}(\theta)$, where $f(\theta)=0$. In this article, we use a theorem due to Kishi and Miyake to give an explicit description of an infinite set ${\mathcal F}$ of cubic polynomials $f(x)\in {\mathbb Z}[x]$, such that $f(x)$ satisfies the three conditions: ${\rm Gal}_{{\mathbb Q}}(f)\simeq S_3$, the class number of the unique quadratic subfield of the splitting field of $f(x)$ is divisible by 3, and $f(x)$ is monogenic. Moreover, if $g(x)$ is any cubic polynomial satisfying these three conditions, then there exists a unique $f(x)\in {\mathcal F}$ such that $f(x)$ and $g(x)$ generate the same cubic extension over ${\mathbb Q}$.