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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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up vote 9 down vote accepted

I haven't read Neukirch's book (yet) so I don't know the context exactly but I would assume that he is referring to the fact that in general you cannot put an order in the set of complex numbers, or in your case the ring of Gaussian integers in a way that the order is compatible with the algebraic operations of addition and multiplication.

The concept is called an ordered ring. Also take a look at the Wikipedia article about ordered fields In particular take a look at the section on which fields can be ordered, where it mentions the case of the complex numbers.

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 $-1 > 0$.

The other case is similar.

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"you cannot have both 1>0 and −1>0." can you elaborate this? our order can be very weird –  Hedgehog Oct 29 at 13:05

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