Re-reading an algebra course I was wondering about something.
Let $R$ be a Euclidean domain endowed with a Euclidean function $f$, ie a function satisfying :
- $f(a) \leq f(ab) \, \,\forall \, a, b \in R\backslash\{0\}$
- $\forall \, (a,b) \in R\times R\backslash\{0\} : \, \exists \, q,r \in R, a = bq + r$ with either $r = 0$ or $f(r) < f(b)$
The latter axiom being Euclidean division (I know the first one is not 100% necessary but that's not my point here).
What I'm wondering is whether this Euclidean division is unique.
No. Consider in $\mathbb{Z}[i]$ (with the complex modulus) two possible divisions of $1+i$ by $2$: $1+i=2*0+(1+i)$, $1+i=2*1+(i-1)$.
I can’t find a reference, but I remember reading that every Euclidean domain where the Euclidean division is unique is either a polynomial ring in one variable over a field or $\mathbb{Z}$.