$\mathbb{C}$ is not an ordered fieldMathbbCIsNotAnOrderedField2006-10-01 00:33:012006-10-07 22:25:22Theoremordered ring\usepackage{amssymb}
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\begin{thm}
$\mathbb{C}$ is not an ordered field.
\end{thm}
First, the following theorem will be proven:
\begin{thm}
$\mathbb{Z}[i]$ is not an ordered ring.
\end{thm}
\begin{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.
\end{proof}
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.