Given $n,m \in \Bbb Z[i],$ we have $n = qm + r,\ r \in \Bbb Z[i].$ Then what is the value of $\displaystyle {\inf\limits_{n,m} \dfrac {\deg (m)} {\deg (r)}}\ ?$
How do I solve this question? Any help will be highly appreciated.
Thanks in advance.
EDIT $:$ How do I define that ratio if $r = 0\ ?$ For instance we could take $n = 1 + i$ and $m = 1 - i.$ Then indeed $r = 0.$ Then $\deg (r) = {\left \lvert r \right \rvert}^2 = 0.$ If $r \neq 0$ then $\deg (r) \lt \deg (m).$ Hence $\dfrac {\deg (m)} {\deg (r)} \gt 1.$ Also I can see that $\deg (r) \leq \dfrac {1} {2} \deg (m).$ So if $r \neq 0$ then $\dfrac {\deg (m)} {\deg (r)} \geq 2$ and if we take $n = 1 + i$ and $m = 2$ then the ratio is exactly equal to $2.$ So in this case the infimum turns out to be $2.$ But what will happen if $r = 0\ $?