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[parent] $\mathbb{C}$ is not an ordered field (Theorem)
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. $ \qedsymbol$

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.



"$\mathbb{C}$ is not an ordered field" is owned by Wkbj79.
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See Also: complex, basic facts about ordered rings


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Cross-references: ring, contradiction, total ordering, basic facts about ordered rings, ordered ring, ordered field
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This is version 7 of $\mathbb{C}$ is not an ordered field, born on 2006-10-01, modified 2006-10-07.
Object id is 8406, canonical name is MathbbCIsNotAnOrderedField.
Accessed 960 times total.

Classification:
AMS MSC12J15 (Field theory and polynomials :: Topological fields :: Ordered fields)
 13J25 (Commutative rings and algebras :: Topological rings and modules :: Ordered rings)
 06F25 (Order, lattices, ordered algebraic structures :: Ordered structures :: Ordered rings, algebras, modules)

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