An Euler phi function for the Eisenstein integers and some applications

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.