An Euler phi function for the Eisenstein integers and some applications

TL;DR

This paper extends the classical Euler phi function to Eisenstein integers, explores the structure of their unit groups, and demonstrates that Euler-Fermat theorem applies in this complex integer setting, with applications to cyclicity criteria.

Contribution

It introduces a generalized Euler phi function for Eisenstein integers and proves its properties, including the applicability of Euler-Fermat theorem and criteria for cyclic unit groups.

Findings

Generalized Euler phi function for Eisenstein integers established.

Proved Euler-Fermat theorem for Eisenstein integers.

Identified conditions for cyclicity of certain unit groups.

Abstract

The Euler phi function on a given integer $n$ yields the number of positive integers less than $n$ that are relatively prime to $n$. Equivalently, it gives the order of the group of units in the quotient ring $\mathbb{Z}/(n)$. We generalize the Euler phi function to the Eisenstein integer ring $\mathbb{Z}[\rho]$ where $\rho$ is the primitive third root of unity $e^{2\pi i/3}$ by finding the order of the group of units in the ring $\mathbb{Z}[\rho]/(\theta)$ for any given Eisenstein integer $\theta$. As one application we investigate a sufficiency criterion for when certain unit groups $\left(\mathbb{Z}[\rho]/(\gamma^n)\right)^\times$ are cyclic where $\gamma$ is prime in $\mathbb{Z}[\rho]$ and $n \in \mathbb{N}$, thereby generalizing well-known results of similar applications in the integers and some lesser known results in the Gaussian integers. As another application, we prove that the celebrated Euler-Fermat theorem holds for the Eisenstein integers.