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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 $\le$ . 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, $-1=i \cdot i \ge 0 \cdot i=0$ , 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 $\mathbb{C}$ is not an ordered field.
How to Cite This Entry
Warren Buck. " is not an ordered field" (version 7). PlanetMath.org. Freely available at http://planetmath.org/MathbbCIsNotAnOrderedField.html.
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