Exercise 1.8

Question

Exercise 8: Prove that no order can be defined in the complex field that turns it into an ordered field.

ghostofgarborg · Accepted Answer

By proposition 1.18d, an ordering $<$ that makes $\mathbb{C}$ an ordered field would have to satisfy
$-1 = i^2 > 0$, contradicting $1 > 0$.

Amit Awasthi · Answer

\begin{proof}
      Assume $\mathbb{C}$ is an ordered field. By Proposition $1.18(d)$, if $x 
e 0$, $x^2 > 0$. By Theorem 1.27, $i 
e 0$, but by Theorem 1.28, $i^2 = -1 < 0$, which is a contradiction.
\end{proof}

Exercise 1.8

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Exercise 1.8

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