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. AxelBoldt 16:41, 18 Jun 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. Pfortuny 20:14, 18 Jun 2004 (UTC)
The PM definition might be enough in some 'one-dimensional' contexts - algebraic number fields, algebraic curves. Things get a bit more serious ... The Soviet encyclopedia seems to agree with WP on this.
Charles Matthews 20:24, 18 Jun 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? AxelBoldt 08:55, 20 Jun 2004 (UTC)
- I've remembered... Let be a valuation with . Then you can define:
- (with ) 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 :) Pfortuny 10:29, 20 Jun 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? - 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!) - Gauge 22:53, 19 Jun 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. Daniele.tampieri 23:07, 1 December 2006 (UTC)
Planetmath disagree with J.P.Serre, A.Weyl and S.Lang and also with Bourbaki ('though the last is manly 25 years old memory). In facts:
- Planetmath defines a norm or an absolute value.
- Defining |x|=e-v(x) defines an equivalence between absolute values and valuations so the two theories are equivalents, with a difference on focus.
- Absolute value satisfies the triangular inequality (|x+y|≤|x|+|y| <=> the side of a triangle is less (or equal) than the sum of the two others <=> the straight line is the shortest path). In that case, the space is metric, meaning |x-y| is a distance measure compatible with the algebraic structure.
- Valuations have a different purpose. They generalize the concept of multiplicity and are also used for classification.
- Valuations satisfies the ultrametric inequality (|x+y|≤max(|x|,|y|) <=> the multiplicty of a sum is not greater than the mutiplicty of each summand). The ultramertic inequality is stronger than the triangular, so the space with a valuation are metric, hence their name "ultrametric".
- In a ultrametric space, if two circles intersects then one contains the other. Therefore, the set of circles |x|<r defines a well behaved classifying tree like the directories in file system on computers).
- A field with a ultrametric absolute value is often called non-archimedean. This causes a little bit of a confusion with archimedean ordered field, that is a field in which every element x is finite, meaning x<n for an integer n.
- The two notions are not equivalent. For example, the field of surreal_number has an archimedean (non-ultrametric) absolute value but is non-archimedean as an ordered field. The absolute value of infinite elements are not real but in an extension of 'R'.
- However, there is a canonical way of building a valuation on a ordered field (c.f. E. Artin, although he do not claim to be the inventor). This valuation is always non-archimedean (ultramatric). And the ordered field is archimedean (has no infinte elements) if and only if the valuation is trivial.