TITLE: Can we make $\mathbb{Z}$ into a field?
QUESTION [8 upvotes]: This is probably an elementary question about fields, but I think it is a little tricky. 

Can we make the integers $\mathbb{Z}$ into a field?

Let me be more precise. Is it possible to make $\mathbb{Z}$ into a field so that the underlying additive group structure is the usual addition in $\mathbb{Z}$? In other words, we just need to define "multiplication" on $\mathbb{Z}$ that makes it into a field. But the question is: How to define such a multiplication?
The ordinary multiplication (that we learn in elementary school) doesn't work as $2$ does not have a multiplicative inverse.

REPLY [5 votes]: Theorem: If $F$ is a field and $x \in F$ with $x \neq 0$, then either:

there exists $y \in F$ such that $y + y = x$, or
$x + x = 0$.