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Created local field. I discovered there’s no number field and no Pontryagin duality.
Created Pontryagin duality.
There’s not much relation. You can say that the -adic local fields have local rings in them: the elements of norm 1 or less, where the maximal ideal is the set of elements of norm less than 1.
In general, given a local ring , it’s possible to complete it by taking an inverse limit of the quotient rings , and the -adic local fields arise as fields of fractions of completions of localizations of rings of integers in number fields. But those local rings coming from number field are a tiny fraction of local rings in general.
Don’t forget to put answers to questions into the corresponding nLab page. I pasted that answer here.
Okay, Urs – thanks.
I’ve also written brief articles on number field and Pontryagin duality. Pages for Haar measure and ring of adeles are on to-do list (but please be my guest, anyone).
Thanks. Added some hyperlins to number field.
Thanks a lot for Pontryagin duality. I also added some hyperlinks here.
Pontrjagin is one of the rare names which are very often spelled with correct transliteration spelling, as in English edition of Russian math journals and in Math Reviews of Russian papers. It is a pity to see it as Pontryagin in nlab, especially as there is an entry Lev Pontrjagin and
THERE IS AN old ENTRY Pontrjagin dual with redirect Pontrjagin duality (I was among contributors to the old entry quite a while ago)
so I do not see a need to have separate entry Pontryagin duality. It would be better to merge all under old entry Pontrjagin dual.
There’s not much relation. (local fields vs local rings)
I disagree with this, the basic example of a local ring is the localization of some ring of functions around a point, meaning infinitesimal information around the point. The same kind of examples are basic examples of local fields: one looks now at power series where one can invert, and they do not converge but have sense only in a formal neighborhood. So geometrically the meaning of local is about the same, although the structural definition is a bit different.
It would be better to merge all under old entry Pontrjagin dual.
Right. Could you do it?
THERE IS AN old ENTRY Pontrjagin dual with redirect Pontrjagin duality (I was among contributors to the old entry quite a while ago) so I do not see a need to have separate entry Pontryagin duality. It would be better to merge all under old entry Pontrjagin dual.
I agree one hundred percent, and would add there should be a redirect since in my experience ’Pontryagin’ is a far more common transliteration. However, there’s no need to shout (caps).
The same kind of examples are basic examples of local fields
No. “Local field” in mathematical English means what I said it means, and you only get them starting from very specific types of local rings. Sure, no one denies there are some thematic commonalities (and I also tried to convey them in my comment), but I interpreted David’s question structurally, as asking whether there’s a functor from local rings, or more to the point local integral domains to local fields, and the answer is no. There is of course a forgetful functor from local fields to local rings (which assigns the maximal compact subring), but the possible values up to isomorphism are a pretty small fraction of local rings or local integral domains.
My sense is that the “local” in “local field” was chosen by Weil (or whoever it was) more with regard to the topological local compactness than to the algebraic sense of local, otherwise it would be highly misleading.
Yes, “local” in “local integral domain” was in the algebraic sense. But Zoran’s answer was wrong: that kind of local is simply not the meaning of “local field” as it is used in mathematical English. (Or, if anyone ever does use it that way, then they are in serious conflict with established use in number theory. Citation?)
I will reiterate that the “local” in “local field” probably refers to local compactness, since that is the relevant structure in the definition!
Right. Could you do it?
Not before next Tuesday. I am at a conference with 3 other deadlines this week and next Monday.
Don’t worry, Urs – I’m happy to do it. Right now I’m at local field, editing to clear up some evident misconceptions. :-)
Okay, I’m done editing for now at local field. I invite those who know what a local field is :-D to have a look to see if they think I got it right.
there’s no need to shout (caps)
According to the major epistemological work of Hirsch, the true interpretation is the intended meaning and not the convention of the particular circle of people who happen to receive it. Hence the shouting interpretation is wrong. It takes more time to type the code for italics than to type capitals.
According to the major epistemological work of Hirsch
Pfft. It would be better, Zoran, if you took the time to put your text between two asterisks (which for me takes less time), and not flout well-known conventions. This is a matter of basic Netiquette. I would say more, except that I am in no mood to waste any more time arguing with you on this.
Merged contents of former article on Pontryagin duality to Pontrjagin dual, and added a redirect.
I agree with Todd #19. Basic politeness involves everyone in a given community using the same conventions and social norms. It has nothing to do with meaning.
Netiquette
Guys, I am sorry. I consider internet just an extension of the usual writing. Consider me impolite if you wish, it is not intended. I prefer interpreting by knowing people and not by colorless conventions and unformeness. True interpretation requires knowing the source; so one who does not know them restrains from hard interpretations.
your text between two asterisks (which for me takes less time)
Sorry but I did not come up with that< it would come automatically in nlab but for nforum I am still not subconsciously ready for such (I assumed em em wiki environment, I am in completely different envirnoment and mindset and your requests are really difficult for somebody in the middle of the conference and who slept 3 and half hours last night after 7 hours in the bus).
I should add that I lost about 10 minutes of concentration after being stressed for shouting which I did not do. I do not feel nice about nforum the whole afternoon after that.
Thanks for your words, Zoran. I guess I should apologize, too, because I sort of did take a few piss shots over the meaning of local field (although it was an intellectual point that needed clarification, I felt).
The problem with the internet is that in most cases we don’t know each other (very well), we’ve never met each other in person, we can’t rely on facial and bodily cues, or interrupt to forestall a misunderstanding, etc. etc. That’s the whole point of abiding by simple rules of netiquette: that writing emails and comments on the internet is decidedly not an extension of ordinary writing! (Or, at least adopting such an attitude is fraught with some obvious pitfalls.)
Thanks for clarifying, though, and I for my part will try to be less reactive. (I need to catch up on sleep too!)
Don’t worry :) and get some sleep…
added the following paragraph to the Idea-section at local field (and will add a corresponding paragraph to global field):
Basic examples are the p-adic numbers and the field of Laurent series over a finite field . Local fields are opposite to global fields in that where (under the function field analogy) the latter may be thought of as fields of rational functions on arithmetic curves, local fields are like fields of functions on formal disks inside such curves. Accordingly the Langlands correspondence for global fields has a “localization” to the local Langlands corrrespondence for local fields.
added pointer to:
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