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

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$\mathbb{C}$ 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 and

Note also that $i \neq 0$ . Thus, either or . 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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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 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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