A Division Algorithm for the Gaussian Integers' Minimal Euclidean Function

2025/02/28 by Graves, Hester
#11A05 #11A63 #11R04 #11R11 #11R99 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2502.21136

Abstract

The usual division algorithms on ℤ and 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 ∖ 0 → ℕ on a Euclidean domain R is itself a Euclidean function, called the minimal Euclidean function and denoted by ϕ_R. The integers, ℤ, and the Gaussians, ℤ[i], are the only rings of integers of number fields for which we have a formula to compute their minimal Euclidean functions, ϕ_ℤ and ϕ_ℤ[i]. This paper presents the first division algorithm for ℤ[i] relative to ϕ_ℤ[i], empowering readers to perform the Euclidean algorithm on ℤ[i] using its minimal Euclidean function.

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