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Math Forum » Discussions » sci.math.* » sci.math.independent

Topic: What is the intuitive meaning of "non-Archimedean" for a valued field?
Replies: 11   Last Post: May 15, 2013 4:29 PM

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Virgil

Posts: 5,489
Registered: 1/6/11
Re: What is the intuitive meaning of "non-Archimedean" for a valued field?
Posted: May 13, 2013 11:17 PM
  Click to see the message monospaced in plain text Plain Text   Click to reply to this topic Reply

In article
<bf23b508-9d6c-459b-b797-5022f1dd0e5d@tz3g2000pbb.googlegroups.com>,
mike3 <mike4ty4@yahoo.com> wrote:

> Hi.
>
> I'm curious about this. The "Archimedean property" for an _ordered_
> field F means that given any positive elements a and b in F, with a <
> b, then there exists a natural number n such that na < b.

Not quite as stated above.

Given 0 < a < b there must be some natural n such that na > b.

But if a is negative, one will have na < b for all naturals n.

The standard ordered field of reals and all of of its subfields have
the property, but fields with infinitesimal elements do not.
--





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