# Euclidean Function at 0

In a few places where I have looked the Euclidean Function of a Euclidean Domain is only being defined for non-zero elements. I am teaching an undergraduate course and I am trying to make things simpler as possible. Is there any good reason why not to define it as $0$ at $0$?

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+1: this certainly answers the question. I found it somewhat unsatisfying though that -- so far as I could see -- Nagata's 1978 example of a Euclidean domain with a transfinite algorithm but no $\mathbb{Z}^+$-valued algorithm is listed in the bibliography but not discussed (or even cited!) in the text at all. Instead they give the example of $\mathbb{Z} \oplus \mathbb{Z}$, which is not a domain, so seems rather cheap. According to MathSciNet, before Nagata, Hiblot gave a similar example (first incorrectly, but corrected before Nagata's paper). –  Pete L. Clark Jul 13 '10 at 22:14