TITLE: Euclidean function of Euclidean domain defined at 0
QUESTION [5 upvotes]: 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 as simple as possible. Is there any good reason why not to define it as $0$ at $0$?

REPLY [3 votes]: Old thread, but nobody wrote the simple practical reason, so I'll do it now. In $K[T]$, we want to use $\deg$ as a Euclidean function, so $\deg(c) = 0$ for $c \in K - \{0\}$. But $\deg(0)$ is best thought of as $-\infty$, which breaks the usual definitions of "Euclidean function". It's common to sidestep this by simply not requiring the Euclidean function to be defined at 0 at all, which is fine since you never need it anyway.