On ideal class groups of totally degenerate number rings

TL;DR

This paper investigates the structure and size of ideal class groups of certain number rings defined by monic polynomials with integer roots, providing formulas, asymptotic analysis, and structural descriptions for low degrees.

Contribution

It introduces formulas for the orders of ideal class groups of rings defined by monic polynomials with integer roots and analyzes their asymptotic behavior, including structural results for degrees 2 and 3.

Findings

Formulas for the orders of ideal class groups and monoids

Asymptotic behavior of class groups as discriminant grows

Structural description of class groups for degree 2 and 3

Abstract

Let $\chi(x)\in \mathbb{Z}[x]$ be a monic polynomial whose roots are distinct integers. We study the ideal class monoid and the ideal class group of the ring $\mathbb{Z}[x]/(\chi(x))$. We obtain formulas for the orders of these objects, and study their asymptotic behavior as the discriminant of $\chi(x)$ tends to infinity, in analogy with the Brauer-Siegel theorem. Finally, we describe the structure of the ideal class group when the degree of $\chi(x)$ is $2$ or $3$.