# Gaussian Integer Cannot be Ordered

I just started to read Neukirch's Algebraic Number Theory book. On page 5 , the book has the following exercise.

Show that the ring $\mathbb{Z}[i]$ cannot be ordered

I don't quite understand with that question. Since $\mathbb{Z}[i]$ is countable, we can enumerate elements of $\mathbb{Z}[i]$ and call the first element on the list as the smallest element and so on. So what is the question really asking here?

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What you usually do to show this is to get a contradiction when you try to see what relation the element $i \in \mathbb{Z}[i]$ should have.
For example if you assume that there is an order in the Gaussian integers, then you should have $i > 0$ or $i < 0$. Then in the first case this implies that $-1 = i^2 > 0$ and then in turn $1 = (-1) \cdot (-1) > 0$ which gives you a contradiction because you cannot have both $1 > 0$ and $-1 > 0$.