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

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

First, the following theorem will be proven:

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

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 $i\eq 0$ . 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.

单词	mathbbCIsNotAnOrderedField
释义	$\\mathbb{C}$ 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 $i\eq 0$ . 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.
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