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Hello Math StackExchange,

The ring of integers $\mathbb{Z}$ is usually endowed with the natural Euclidean function $d(x) = |x|$, making it a Euclidean domain. My question is: Are there any other Euclidean functions that $\mathbb{Z}$ can be endowed with?

What about $\mathbb{Z}[i]$, the Gaussian integers?

Thank you so much!

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Yes, there are several possible Euclidean functions for $\mathbb{Z}$. For example, $d_1(x)=|x|$, or $d_2(x)=\lceil \log_2 (|x|)\rceil $, or $d_3(x)$ defined by $|x|$ for $x\neq 5$ and $d_3(5)=13$. We have a lot of freedom here. Of course, the same applies for other Euclidean rings, like $\mathbb{Z}[i]$. Usually we take the norm $N(z)=z\cdot \overline{z}$. However, some rings are Euclidean but not with the norm function (they are not norm-Euclidean). A famous example is the ring $\mathbb{Z}[\sqrt{14}]$.

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  • $\begingroup$ Hmm, thank you. I found some such functions myself (e.g. in $\mathbb{Z}$, the largest prime-power to divide the input) and realized that there were probably many, as you implied. But many of the elementary results of Euclidean domains (e.g. being principal ideal rings and unique factorization domains, or what the prime elements turn out to be, or what GCDs are) are independent of our choice of Euclidean function. So it seems like a bad question to me. $\endgroup$ – Dfrtbx Apr 21 '14 at 22:20
  • $\begingroup$ I think the question is not so bad. Recently there have been several new results in this direction. $\endgroup$ – Dietrich Burde Apr 22 '14 at 9:07

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