Search All of the Math Forum:

Views expressed in these public forums are not endorsed by NCTM or The Math Forum.

Notice: We are no longer accepting new posts, but the forums will continue to be readable.

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

 Messages: [ Previous | Next ]
 William Elliot Posts: 2,637 Registered: 1/8/12
Re: What is the intuitive meaning of "non-Archimedean" for a valued
field?

Posted: May 13, 2013 10:27 PM

On Mon, 13 May 2013, mike3 wrote:

> 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.

> 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?

> 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.

Date Subject Author
5/13/13 mike3
5/13/13 William Elliot
5/13/13 mike3
5/13/13 Virgil
5/13/13 Virgil
5/13/13 mike3
5/14/13 Virgil
5/14/13 mike3
5/14/13 Virgil
5/14/13 Brian Q. Hutchings
5/15/13 Brian Q. Hutchings
5/14/13 FredJeffries@gmail.com