Is the Euclidean valuation on a Euclidean domain unique? I have just started studying Euclidean domain which is defined as follows :
An integral domain $R$ is called a Euclidean domain if there exists a function $\delta : R \setminus {\{0\}} \longrightarrow \mathbb {N_{0}}$ which satisfies the following properties.
(1) $\delta (a) \leq \delta (ab)$, for all $a,b \in R \setminus {\{0\}}$.
(2) For any $a,b \in R$ with $b \neq 0$ there exist $q,r \in R$ such that $a = bq + r$, where either $r = 0$ or $\delta (r) < \delta (b)$.
Here $q$ is called quotient and $r$ is called remainder.
The function $\delta$ is called a Euclidean norm function (or Euclidean valuation) on $R$.
Now my question is :
Is such Euclidean valuation on a Euclidean domain unique?
I am in a fix.Please tell me.
Thank you in advance.

Another example (to show non-uniqueness): On the ring of integers, define $\delta (x) = x^2$.
