A Division Algorithm for the Gaussian Integers' Minimal Euclidean Function
Fuente:
arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866916635460239360 |
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| author | Graves, Hester |
| author_facet | Graves, Hester |
| contents | 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 $ϕ_R$. 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, $ϕ_{\mathbb{Z}}$ and $ϕ_{\mathbb{Z}[i]}$. This paper presents the first division algorithm for $\mathbb{Z}[i]$ relative to $ϕ_{\mathbb{Z}[i]}$, empowering readers to perform the Euclidean algorithm on $\mathbb{Z}[i]$ using its minimal Euclidean function. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_21136 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Division Algorithm for the Gaussian Integers' Minimal Euclidean Function Graves, Hester Number Theory 11A05, 11A63, 11R04, 11R11, 11R99 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 $ϕ_R$. 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, $ϕ_{\mathbb{Z}}$ and $ϕ_{\mathbb{Z}[i]}$. This paper presents the first division algorithm for $\mathbb{Z}[i]$ relative to $ϕ_{\mathbb{Z}[i]}$, empowering readers to perform the Euclidean algorithm on $\mathbb{Z}[i]$ using its minimal Euclidean function. |
| title | A Division Algorithm for the Gaussian Integers' Minimal Euclidean Function |
| topic | Number Theory 11A05, 11A63, 11R04, 11R11, 11R99 |
| url | https://arxiv.org/abs/2502.21136 |