A Division Algorithm for the Gaussian Integers' Minimal Euclidean Function

Abstract

The usual division algorithms on $\mathbb{Z}$ and $\mathbb{Z}[i]$ measure the size of remainders using the norm function. These rings are Euclidean with respect to several functions. The pointwise minimum of all Euclidean functions $f: R \setminus 0 \rightarrow \mathbb{N}$ on a Euclidean domain $R$ is itself a Euclidean function, called the minimal Euclidean function and denoted by $\phi_R$. The integers, $\mathbb{Z}$, and the Gaussians, $\mathbb{Z}[i]$, are the only rings of integers of number fields for which we have a formula to compute their minimal Euclidean functions, $\phi_{\mathbb{Z}}$ and $\phi_{\mathbb{Z}[i]}$. This paper presents the first division algorithm for $\mathbb{Z}[i]$ relative to $\phi_{\mathbb{Z}[i]}$, empowering readers to perform the Euclidean algorithm on $\mathbb{Z}[i]$ using its minimal Euclidean function.