Multiplicatively monotone Euclidean norm admits unique Euclidean division for exact divisor
Suppose for . Then, we cannot write:
with and .
Given: An integral domain with multiplicatively monotone Euclidean norm . and .
To prove: We cannot write with .
Proof: Suppose with . Then, since , we get:
Take both sides, to get:
Since is multiplicatively monotone, we get:
On the other hand, we have, by assumption:
This is a contradiction, completing the proof.