Counting Ideals in $\mathbb{Z}[t]/(f)$

TL;DR

This paper investigates the growth of ideals in quotient rings formed by monic cubic polynomials over integers and computes the ideal zeta function for polynomial rings modulo powers of t, advancing understanding of ideal enumeration.

Contribution

It introduces new methods to analyze ideal growth in specific polynomial quotient rings and computes their ideal zeta functions, providing explicit formulas for these cases.

Findings

Derived the ideal zeta function for $Z[t]/(t^n)$ for all n

Analyzed the growth behavior of ideals in $Z[t]/(f)$ for monic cubic f

Provided explicit computations and formulas for ideal enumeration

Abstract

In this paper we study the growth of ideals in $\mathbb{Z}[t]/(f)$ for a monic cubic polynomial $f$. We also compute the ideal zeta function of $\mathbb{Z}[t]/(t^n)$ for any $n \in \mathbb{N}$.