$ℂ$ 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.

Title	$ℂ$ is not an ordered field
Canonical name	mathbbCIsNotAnOrderedField
Date of creation	2013-03-22 16:17:25
Last modified on	2013-03-22 16:17:25
Owner	Wkbj79 (1863)
Last modified by	Wkbj79 (1863)
Numerical id	10
Author	Wkbj79 (1863)
Entry type	Theorem
Classification	msc 06F25
Classification	msc 13J25
Classification	msc 12J15
Related topic	Complex
Related topic	BasicFactsAboutOrderedRings