Talk:Valuation (algebra)

message: 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.
AlainD (talk) 18:36, 8 February 2007 (UTC)
thread_title:  Clean up 

message: # Regarding the pi-adic valuation in the examples, do we really need the condition that R is a principal ideal domain? Shouldn't a unique factorization domain be sufficient?  — Preceding unsigned comment added by 131.246.164.32 (talk) 10:22, 29 November 2011 (UTC)
thread_title:  Questions 

message: I have removed said section due to errors. The described function was not a valuation, e.g. v_h(h)=\infty. The correct way to do what was intended is via a rank 2 valuation. I might write a corrected version if I get around to it.
Sloth sisyphos (talk) 11:35, 14 December 2021 (UTC)
thread_title:  Section "geometric notion of contact" 

message: An editor has identified a potential problem with the redirect :Krull valuation and has thus listed it for discussion. This discussion is at Wikipedia:Redirects for discussion/Log/2022 July 12#Krull valuation until a consensus is reached, and readers of this page are welcome to contribute to the discussion. Jay (talk) 16:27, 18 July 2022 (UTC)
replies: text: The discussion has been relisted at Wikipedia:Redirects for discussion/Log/2022 July 18#Krull valuation. D.Lazard (talk) 21:12, 18 July 2022 (UTC)
thread_title:  "Krull valuation" listed at [[Wikipedia:Redirects for discussion|Redirects for discussion]]