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![[parent]](http://images.planetmath.org:8080/images/uparrow.png)
 is not an ordered field
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(Theorem)
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First, the following theorem will be proven:
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.
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" is not an ordered field" is owned by Wkbj79.
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Cross-references: ring, contradiction, total ordering, basic facts about ordered rings, ordered ring, theorem, ordered field
There is 1 reference to this entry.
This is version 7 of  is not an ordered field, born on 2006-10-01, modified 2006-10-07.
Object id is 8406, canonical name is MathbbCIsNotAnOrderedField.
Accessed 1520 times total.
Classification:
| AMS MSC: | 12J15 (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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Pending Errata and Addenda
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