Editing Talk:Valuation (algebra) (section)

== Clean up ==
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_(mathematics)|norm]] or an [[absolute value#Fields and rings|absolute value]].
# Defining |x|=e<sup>-v(x)</sup> 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 (mathematics)|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 field|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.
[[User:AlainD|AlainD]] ([[User talk:AlainD|talk]]) 18:36, 8 February 2007 (UTC)