On ideal class groups of totally degenerate number rings

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 $3$.