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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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mike3

Posts: 2,396
Registered: 12/8/04
Re: What is the intuitive meaning of "non-Archimedean" for a valued field?
Posted: May 13, 2013 10:51 PM
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On May 13, 7:27 pm, William Elliot <ma...@panix.com> wrote:
> On Mon, 13 May 2013, mike3 wrote:
> > 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.

>
> That is trival.  If a < b, then 1a < b and the Archimedean property
> always holds.
>


Oops. That should be "na > b". :)

> > Intuitively, this means F has no "infinitely big" or "infinitely small"
> > elements.
> > We could also say that "given any positive element a, then
> > there is a natural number n such that na > 1". If the property fails,
> > then F contains "infinitely small" elements.

>
> What does infinitely large and infinitely small mean?
>


Usually, "infinitely big" means "bigger than any finite number". If
one thinks
of the elements 1, 1+1, 1+1+1, etc. as the finite numbers in the
field, then
if a > 1, a > 1+1, ..., a is "infinitely big". The inverse of an
infinitely big
element is infinitely small.

> > Now, there is an analogous property for non-ordered, "valued" fields
> > (fields with an "absolute value" function added). The "Archimedean
> > property" here means that given any nonzero element a e F, that there
> > exists a natural number n such that |na| > 1. But what, intuitively,
> > does it mean when this property fails?

>
> That there's some element a, for which the sequence
> (|na|)_n is bounded by 1.
>

> > In that case, there aren't any elements with "infinitely small but
> > non-zero" absolute value since the absolute value functions are usually
> > taken as real-valued, and the reals are Archimedean (as an ordered
> > field). Instead, what happens for such a real-valued absolute value is
> > that the triangle inequality strengthens to |a + b| <= max(|a|, |b|) and
> > not just |a + b| <= |a| + | b|. This causes the space to behave really
> > weirdly(*). But what is the _intuition_ here, and how does this notion
> > relate, if at all, to the first one? Especially considering I see in
> > papers like this:

>
> It seems such a norm would give rise to an ultra metric.





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