Talk:Local field

message: I think this article could be improved a great deal.  (I'm a professional pure mathematician, but don't know about local fields.)  The main problem is that the page doesn't seem to contain an actual definition of local field.  
Maybe the first sentence of the article is intended as a definition?  If so, it's certainly not clear that that's the intention: it could just mean that every local field is a locally compact topological field with respect to a non-discrete topology, without that being a sufficient condition.  By way of comparison, an article on topological groups might start "A topological group is a topological space that also carries a group structure", although that's not a sufficient condition (as the group operations have to be continuous).  It's also not usual for Wikipedia mathematics articles to give the definition in the first sentence.
Somewhere in the article should be a sentence beginning: "Definition: a local field is ..."
Another point that needs clarification is whether being local is a property of a field, or extra structure on a field.  The fact that a local field carries a topology suggests that it might be extra structure, so that a field could be a local field in more than one way (just as many groups can be given the structure of a topological group in more than one way).  But perhaps it's a theorem that any given field can be a local field in at most one way.  I don't know: I couldn't tell from the article.  
It would be great if someone with relevant expertise could edit the page to clarify these points.  — Preceding unsigned comment added by 81.109.231.166 (talk) 20:18, 18 August 2012 (UTC)
thread_title:  Where is the definition? 

message: The Introduction includes this as its first sentence:
"In mathematics, a local field is a special type of field that is a locally compact topological field with respect to a non-discrete topology."
This only states that a local field is a special type of field that also has a certain property.  This is not a definition, even if it was intended to be.
This informal description is fine for the Introduction, but there should be a rigorous definition that doesn't merely state that a local field is a certain kind of special type of field!  
Preferably in the first section after the Introduction.
To make matters worse, at the end of the introduction, this passage occurs:
"There is an equivalent definition of non-archimedean local field: it is a field that is complete with respect to a discrete valuation and whose residue field is finite. However, some authors consider a more general notion, requiring only that the residue field be perfect, not necessarily finite. This article uses the former definition."
The phrase "equivalent definition" implies that someone believes a definition has already been presented. But it has not.
To make matters much worse, the phrase "former definition" is included in this passage, even after two definitions are included beyond the nonexistent first definition, making three definitions or nonexistent definitions that have been referred to!  Which one is the "former" definition out of these three?  Much confusion is sowed by this mess!Daqu (talk) 20:15, 21 March 2016 (UTC)
thread_title:  Needs a definition! 

message: Hey all, I know this issue has been discussed above in 2014, but since the corresponding passge is still in the article, I would like to repeat it.
In the case of K=Complex Numbers, the "absolute value" defined here is the square of the usual absolute value and as the example above (
(2+1)^2=9 > 4+1=2^2+1) shows, this does not satisfy the triangle inequalitiy. For all other local fields, the formula should be fine (for R we get the usual absolute value and for non-archimedian fields we have an "ultra-triangle inequality" which should also work).
I came across this issue while working on the German version of this article. I wanted to include this formula from this article only to realize it is false.
Any ideas how to fix this problem?
Best regards, --CoTangent (talk) 11:02, 22 March 2020 (UTC)
thread_title:  This absolute value does not satisfy the triangle inequality in the case of C 

message: The definition given in the first sentence, "a field K is called a local field if it is complete with respect to a topology induced by a discrete valuation v and if its residue field k is finite" is in contradiction with the rest of the paragraph, in that it excludes the Archimedean local fields that are prominently mentioned in what follows. It is true that that definition can be found in the cited reference (Cassels and Fröhlich, p.129), but the author of that article (J.-P. Serre) does not seem to consider the Archimedean case to fall under the "local field" umbrella, as is clear from the classification that follows. (Amusing side note: I tried to locate a definition of local field in Serre's book "Corps locaux"; not only could I not find one (with modest effort), but I also got the impression that the phrase "corps local" actually does not appear at all in that book, apart from in its title.) So it might be true the the real and complex numbers are the least loved among the local fields, or more accurately the fields that care least about the love the get from those studying local fields, but I think this blatant inconsistency should be mended one way or another. Marc van Leeuwen (talk) 14:35, 22 November 2021 (UTC)
thread_title:  Inconsistency in the lead