# $\mathbb{C}$ is not an ordered field

###### Theorem 1.

$\mathbb{C}$ 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 $\leq$. Note that $0<1$ and $-1=-1+0<-1+1=0.$

Note also that $i\neq 0$. Thus, either $i>0$ or $i<0$. In either case, $-1=i\cdot i\geq 0\cdot i=0$, 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 $\mathbb{C}$ is not an ordered field.

Title $\mathbb{C}$ is not an ordered field mathbbCIsNotAnOrderedField 2013-03-22 16:17:25 2013-03-22 16:17:25 Wkbj79 (1863) Wkbj79 (1863) 10 Wkbj79 (1863) Theorem msc 06F25 msc 13J25 msc 12J15 Complex BasicFactsAboutOrderedRings