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Definition of ordered field

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Herb

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Feb 8, 2005, 6:11:20 AM2/8/05
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My book states Definition of ordered field as follows:
If a field F has nonempty subset S satisfying
(i) For all x,y in S, x+y in S, x*y in S
(ii) For all x in F, x in S or x=0 or x in -S
Then, F is an ordered field.

Is this definition correct?
At this definition, I wonder wheather S can be F or not.

If S can be F, then
every field satisfies above definition(by seeing S as F)
hence, I reach every field is ordered field.
But I think this is not true.
So, I'm confused.

If someone has precise definition,
please post reply.

Thanks in advance.


William Elliot

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Feb 8, 2005, 7:11:40 AM2/8/05
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On Tue, 8 Feb 2005, Herb wrote:

> My book states Definition of ordered field as follows:
> If a field F has nonempty subset S satisfying
> (i) For all x,y in S, x+y in S, x*y in S
> (ii) For all x in F, x in S or x=0 or x in -S
> Then, F is an ordered field.
>
> Is this definition correct?

> At this definition, I wonder wheather S can be F or not.
>
> If S can be F, then
> every field satisfies above definition(by seeing S as F)
> hence, I reach every field is ordered field.
> But I think this is not true.
> So, I'm confused.
>

It would have the trivial order. Namely x <= y iff x = y

> If someone has precise definition, please post reply.
>

Add axiom 0 not in S.

Brian M. Scott

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Feb 8, 2005, 9:52:31 AM2/8/05
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On Tue, 8 Feb 2005 20:11:20 +0900, Herb <fuzz...@nate.com>
wrote in <news:cua663$nc4$1...@news1.kornet.net> in
alt.math.undergrad:

> My book states Definition of ordered field as follows:
> If a field F has nonempty subset S satisfying
> (i) For all x,y in S, x+y in S, x*y in S
> (ii) For all x in F, x in S or x=0 or x in -S
> Then, F is an ordered field.

> Is this definition correct?

Not quite: (ii) should be

(ii') For all x in F, exactly one of the following holds:
x in S, x = 0, or x in -S.

[...]

Brian

Todd Trimble

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Feb 8, 2005, 10:23:29 AM2/8/05
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On Tue, 8 Feb 2005 20:11:20 +0900, Herb wrote:
>My book states Definition of ordered field as follows:
>If a field F has nonempty subset S satisfying
>(i) For all x,y in S, x+y in S, x*y in S
>(ii) For all x in F, x in S or x=0 or x in -S

Better to say (ii) For all x in F, exactly one of the following
holds: x in S, x = 0, -x in S. (Then S can't be F for instance.)

>Then, F is an ordered field.
>
>Is this definition correct?
>At this definition, I wonder wheather S can be F or not.
>
>If S can be F, then
>every field satisfies above definition(by seeing S as F)
>hence, I reach every field is ordered field.
>But I think this is not true.
>So, I'm confused.
>
>If someone has precise definition,
>please post reply.
>
>Thanks in advance.

Todd Trimble

William Elliot

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Feb 9, 2005, 2:38:51 AM2/9/05
to

(ii') iff (ii) and 0 not in S

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