Editing Talk:Valuation (algebra) (section)

==Untitled==
The concept described in http://planetmath.org/encyclopedia/Valuation.html seems to be closely related, but I think neither is a special case of the other. Is the PlanetMath version used much, should we mention it? They claim that their valuations are used to define general primes in arbitrary fields. [[User:AxelBoldt|AxelBoldt]] 16:41, 18 June 2004 (UTC)

:PlanetMath is not defining the standard concept of valuation, but one I have stumbled upon some times with which I am not quite familiar and which I ''tend to distrust'' as it includes only a subset of what we call here "valuations" (and includes any kind of ''metric''). I am not aware whether in fact what PMath calls ''valuation'' is strictly speaking known as ''valuation'' generally. What they call "non-archimedian" valuations are a small family of valuations in "our" sense. It may be worth mentioning it, but I am not an expert ''in that part of the concept''.

:Just in case: the fact that ''I'' have an opinion shows only that I have one, as any other contributor. [[User:Pfortuny|Pfortuny]] 20:14, 18 June 2004 (UTC)

::The PM definition might be enough in some 'one-dimensional' contexts - [[algebraic number field]]s, [[algebraic curve]]s. Things get a bit more serious ... The Soviet encyclopedia seems to agree with WP on this.

::[[User:Charles Matthews|Charles Matthews]] 20:24, 18 June 2004 (UTC)

:::Let's go with the Soviets then :-) I think the PM contributor is an algebraic number field kind of guy, and in that context the two concepts are probably closely related if not equivalent. Presumably, they like to say that the reals and the ''p''-adic numbers are the completions of '''Q''' with respect to all its valuations.

:::Our valuations do not seem to turn the field into a topological field though, do they? [[User:AxelBoldt|AxelBoldt]] 08:55, 20 June 2004 (UTC)

::::I've remembered... Let <math>\nu:K\rightarrow \Gamma</math> be a valuation with <math>\Gamma\subset\mathbb{R}</math>. Then you can define:

:::::<math>\vert x \vert = \frac{1}{2^{\nu(x)}}</math>

::::(with <math>\vert 0 \vert := 0</math>) and get a ''nonarchimedian valuation'' (if I recall well it is always nonarchimedian) in PM's sense. But you lose the richness of valuations with $\Gamma$ very big (for example, most valuations on function fields of varieties of dimension greater than 1). 

::::You get the topological field using the distance induced by that ''modulus''. Easily, you ''cannot'' do that if the value group is not a subset of the reals.

::::I hope I am not completely misled :) [[User:Pfortuny|Pfortuny]] 10:29, 20 June 2004 (UTC)

:::::I've attempted to introduce the notions of equivalence and "places" here, but the reference that I was using only referred to the number field case, so take it with a grain of salt.  What is the correct definition of equivalence of valuations that includes both group and number field cases? - [[User:Gauge|Gauge]] 07:33, 15 May 2005 (UTC)

::::::The code was looking horrendous, so I cleaned it up a bit.  From what I understand of wikipedia standards, only function and variable names should be italicized (in particular, parentheses shouldn't!)  - [[User:Gauge|Gauge]] 22:53, 19 June 2005 (UTC)

:::::::I have started an extensive clean up of this voice, but I am not an algebraist, nor a logician: the contents surely need a check. [[User:Daniele.tampieri|Daniele.tampieri]] 23:07, 1 December 2006 (UTC)