$ℂ$ is not an ordered field

Theorem 1.

$ℂ$ is not an ordered field.

First, the following theorem will be proven:

Theorem 2.

$\mathbb{Z}[i]$ is not an ordered ring.

Proof.

Many facts that are used here are proven in the entry regarding basic facts about ordered rings.

Suppose that $\mathbb{Z}[i]$ is an ordered ring under some total ordering $\le$. Note that and

Note also that $i\ne 0$. Thus, either $i>0$ or . In either case, $-1=i\cdot i\ge 0\cdot i=0$, a contradiction.

It follows that $\mathbb{Z}[i]$ is not an ordered ring.∎

Because of theorem 2, no ring containing $\mathbb{Z}[i]$ can be an ordered ring. It follows that $ℂ$ is not an ordered field.