I was reading about examples of Euclidean domains and their proofs. I encountered one problem on how to define size function for various Euclidean domains. For example
in $\mathbb{Z}[i]$ size function will be $a^2+b^2$
Similarly for $\mathbb{Z}[(-2)^{1/2}]$ will be $a^2+2b^2$ why can't I define it as $a^2+b^2$ because it will also be from $R-\{0\} \rightarrow N $.
Thanks in advance
You want your size function to be multiplicative, so you need something like $$a+kb\to (a+kb)(a-kb)$$ where $k=\sqrt{-2}$ or whatever.
Example: Set $x=1+\sqrt{-2}, y=1-\sqrt{-2}$.
Your size function gives $|x|=1^2+1^2=2, |y|=1^2+(-1)^2=2$. However $xy=1-(-2)=3$, so your size function gives $|xy|=3^2=9$. This is not multiplicative, since $2\cdot 2\neq 9$.
The standard size function gives $|x|=1^2+2(1^2)=3=|y|$, and $|xy|=9$. Note that $3\cdot 3=9$, as desired.