# Talk:Algebraic number theory

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Field: Number theory

## User:86.128.141.126

This needs to be corrected: "studying them modulo p for all primes p (see finite fields). This is called localization " 86.128.141.126 12:36, 4 March 2007 (UTC)

"Algebraic number theory" can also be interpreted as the arithmetic theory of algebraic numbers. Lhf 11:23, 11 October 2007 (UTC)

## Organization of the article

Since it seems this may be the current COTM, I thought it would be good to suggest first a general structure to the article that could help frame future discussions. Here are some sections I can think of that would help organize the development of the article (I make no claim that this is complete or correct, please suggest additions/modifications):

• History
• Basic notions
• Subfields (aka Branches)
• Major results
• Open questions
• Relation to other fields

Thoughts? RobHar (talk) 23:46, 21 July 2008 (UTC)

## really?

The article currently asserts, more or less, that algebraic number theory is the study of algebraic numbers. I thought it was more number theory (that is, the study of the naturals) using algebraic methods, as distinguished from analytic number theory. Of course the two things might be close to coextensive in practice, but we ought to get the concept right, as currently understood by practitioners in the field. (Clarification: I am not one of those practitioners.) --Trovatore (talk) 04:25, 24 July 2008 (UTC)

It can be taken in both senses, but I believe it's much more common to mean the theory of algebraic numbers (there is, for example, "Transcendental number theory"). Certain older texts on algebraic number theory use "theory of algebraic numbers" in their title (such as Hecke's classic "Lectures on the Theory of Algebraic Numbers", and Ribenboim's "Classical Theory of Algebraic Numbers", both of whose content is pretty standard for an "algebraic number theory" book). Another example is the preface of Neukirch's "algebraic number theory" which consistently refers to the content of the book as being about the study of algebraic numbers. This meaning also appears to be the meaning that springer's encyclopedia of mathematics and planetmath both take. A lot of analytic number theory consists of analytic results about algebraic numbers. And a lot of algebraic number theory uses analytic methods such as automorphic forms, p-adic analysis, p-adic functional analysis to name a few. I think algebraic number theory is defined by the problems it seeks to answer rather than by the methods it uses to answer them, is perhaps a good way to put it. RobHar (talk) 06:22, 24 July 2008 (UTC)
I'm really not sure, and I might myself be an algebraic number theorist (I'm so unsure that I'm unsure about that too!). To my mind, "the theory of algebraic numbers" is a big part of algebraic number theory but not necessarily synonymous with it. Were I to have to give a one sentence definition, I might try: "Algebraic number theory is the study of local and global fields, their associated rings of integers, and other associated structures and invariants, especially the class groups, unit groups and Galois groups." Of course that's much harder for the reader who does not already know some algebraic number theory to understand.
It may not be necessary or profitable to give absolutely the right definition of algebraic number theory before working on the article. I think that Rob (Hi, Rob!) Harron's outline is a good step 0; step 1 (the hard part!) will be to fill in some of these things. I'm not sure yet whether I will participate in this or not -- it might be too ambitious for me. Plclark (talk) 19:53, 25 July 2008 (UTC)Plclark
That is you isn't it? Anyway, I was actually in the process of altering the intro while you were leaving this message, and I think I may have been able to provide a vague enough description for it to be pretty much true, but not too vague as to be completely useless. I said it's a "branch of number theory which studies the algebraic structures related to algebraic integers". I definitely agree that this should only be a first step, and that the description should develop as more work is put in and more people contribute. And if you are who I think you are, plclark, then ambition in writing is right up your alley and you should definitely participate (at least a bit...) RobHar (talk) 20:21, 25 July 2008 (UTC)

## References

I find the statement about the "only classic text still in print [...]" to be somewhat strange. Books like Neukirch's Algebraic Number Theory (ISBN 3540653996) come to mind, and is still available. Particular topics missing? Moreover, the reference sections should not be limited to books currently in print. Stca74 (talk) 15:20, 5 August 2008 (UTC)

Indeed, I believe that section was just pretty much copied over from the analytic number theory article by someone, then I made some changes to make it more relevant, but I do think the reference section as it is should go. But I figured it should be left there until I was (or somebody) going to replace it with another section of references. Perhaps for now I'll quickly adjust it to make it better. But I would hope that would only be a temporary measure as this is currently the (much underworked) collaboration of the month. Though I do think the "only classic" comment is indeed strange for a reference section, it is pretty much accurate (though one could cite Weil's "Basic number theory"). Lang's book was originally published in 1970 as an expansion of a 1964 book, whereas Neukirch's original german edition was published in 1992.
If you have any suggestions of good books that could be included in a nice references (or further reading) section, that would be nice. I'm mainly aware of graduate level texts. Thanks for the prod. RobHar (talk) 17:43, 5 August 2008 (UTC)
The books I thought about and have used most (Fröhlich-Taylor, Neukirch) are now already listed. Perhaps could add Serre's Corps locaux, even though its focus is obviously on the local fields side of things. And I suppose Weil's Basic Number Theory could deserve a mention, although the harmonic analysis approach takes one in slightly different direction. Stca74 (talk) 18:57, 9 August 2008 (UTC)

## Recent content

Hi. I'm not sure adding a bunch of condensed versions of basic articles is the way to go with this article. The inclusion of the section "algebraic integer" is understandable, but basic notions such as "Extension fields" and "Minimal polynomials" are not specific to algebraic number theory, and simply clutter the "Basic notions" section. The "Basic notions" section should be about basic notions of algebraic number theory, not basic notions of field theory, or ring theory. Thoughts? RobHar (talk) 20:57, 23 October 2008 (UTC)

My rationale for adding these new sections is that they help to make the article more accessible. This is a second-level mathematics article, linked directly from number theory and from Portal:Mathematics. I believe the article should aim to give the reader a high-level overview of algebraic number theory, with examples and links to more detailed articles. It needs a lot of work before it can meet that goal. Many readers will not have come here via ring theory or field theory, so they will need a summary of basic concepts such as field extensions, minimal polynomials, algebraic integers, norms and units. Before I started adding sections, the first section under "Basic notions" started with the sentence:
"One of the first properties of Z that can fail in the ring of integers O of an algebraic number field K is that of the unique factorization of integers into prime numbers."
For the general reader who has come here straight from number theory, I don't believe this sentence will make any sense at all - that's why the article needs to start with simpler concepts, building up to factorisation in algebraic number fields and classes of ideals. Anyway, that's my 2c - let's see what others think. Gandalf61 (talk) 09:50, 24 October 2008 (UTC)
I have moved the subsections you created to a new section "Prerequisite notions", so as to avoid confusion.
I guess I don't feel like I should be able to click on a wikipedia article and understand it without clicking the inline links of concepts I don't understand. It is indeed nice to have some introductory material (which the basic notions section already had), and I guess we disagree as to how far back the introductory material has to reach (given how much of it there would need to be). The article as it is is far from complete, and will grow to be quite long, so I feel like adding all this material to it is overkill (the size of the article has almost doubled yet no really new content has been added the encyclopedia). Perhaps, we should consider making an "Introduction to..." article. RobHar (talk) 16:35, 25 October 2008 (UTC)
I am not yet convinced that we need an "Introduction to ..." article in addition to this overview article. However, I will put a note on Wikipedia talk:WikiProject Mathematics to see if other editors have views on this. Gandalf61 (talk) 10:26, 26 October 2008 (UTC)

I tend to agree with Rob. To draw an appropriate image of the domain, it is necessary to make clear what topics the theory builds on. This does not mean, however, that there are several little "stub-sub-articles" put here. It is challenging, but doable (I believe) to roughly outline the majority of notions in one or two sentences - simply take the first sentence of a (hopefully meaningful) lead section of the article, enriched with the relationship to the use etc. of the notion in this article. Often an example is a good way to achieve this, along with (if necessary) the remark that this example is/is not the only source of complication compared to a situation the reader already knows. This should be done in a somewhat inviting way so that the reader feels urged to click at the blue link. So, for example the quote above could be reworded to something like

"Any integer can be uniquely decomposed (or "factored") as a product of prime numbers. The corresponding statement fails for O_K, as the example ... shows. O_K is therefore not a UFD, which yields its treatment more complicated than Z. The class group is a measure for this complication."

I would completely eliminate the whole talk about units at that point. The point that there is the ambiguity in the unit factor is something that IMO clearly belongs to a subarticle. Otherwise you get just too much, as Rob points out. Drawing an image of Z[sqrt(-5)] would be another good way to enlighten the reader. Jakob.scholbach (talk) 11:46, 26 October 2008 (UTC)

It seems to that except for the material on unique factorization and places, the article is not coherent. If I knew lots of abstract algebra but had somehow never encountered a number field, then after reading the article I still wouldn't have a good grasp of what makes number fields interesting. I think this is closely related to the disagreement over the level of the article: If the article had a focused presentation, then it would be easy to decide whether to write "Introduction to ..." or not. As it is, it's hard to tell whether or not that will be useful because it's hard to tell what the article is about.
As always, this article should start with the history. Where did algebraic number theory come from, after all, but the study of the integers? If we explain how to get from Pell's equation and Fermat's Last Theorem to number fields, then number fields will look useful and interesting; then all of the machinery that comes with them will look interesting, too. There should be a discussion of quadratic reciprocity and other reciprocity laws, eventually leading to Artin reciprocity and its requisite machinery. And so on, piggybacking the motivation for sections of the article on the motivations of the great algebraic number theorists.
An aside: The article doesn't mention function fields at all. That'll have to be fixed at some point. Ozob (talk) 22:51, 26 October 2008 (UTC)
Just a quick comment here. The article is indeed far far far from done. It was made collaboration of the "month" in mid-july and pretty much no one has done anything (except mostly me) since then. Here's what it looked like before [1]. There is a lot to be added to the article, most notably history. But the discussion at hand is whether the recent additions are what is needed. RobHar (talk) 23:03, 26 October 2008 (UTC)
Okay. Consensus is obviously solidly against the new content I have added, so I have reverted the article to the October 18 version immediately before I started my additions. I am done here. Gandalf61 (talk) 09:27, 27 October 2008 (UTC)
carl-m pointed me at this article after I asked for something to help with. Anyway As you guys probably want some input from someone that is a "newb" to the type of math I fit that description. I'm a math major at a public university in the US, but I have not yet taken a course about this topic.
1. What strikes me immediately in the intro is mentioning topics that I have never heard of. "This is generally accomplished by considering a ring of algebraic integers O in an algebraic number field K/Q (i.e. a finite extension of the rational numbers Q), and studying the properties of these rings and fields (e.g. factorization, ideals, field extensions)." If we are going to mention these... and we aim to target the general reader (one that has no prior knowledge of this topic) perhaps we could be let in a little more easily?
2. There is no history of the topic, this with my level of knowledge (post calculas, some other unrelated courses after calc) I'm sure I can research up something of the history if you guys wish.
3. The rest of the article suffers from no easy intro or idea of what this is useful for. You have to understand what a "ring" of numbers is before you can access this article comfortably. If you guys are willing to solve the prior problems, I can try to help you with making the first parts of it accessible. (if that is your interest). I'll check back in a few days. —— nixeagle 03:05, 29 October 2008 (UTC)
Hey nixeagle. Indeed this article is lacking in several respects including its accessibility. The intro certainly needs to be rewritten and expanded, and a history section is strongly lacking. I have tried to focus my own efforts on getting certain concepts and facts up, but have not had the time to finish that, or move on to making the article accessible, and I haven't yet attempted to make a history section (though I really want one). Your help with any of this would be great! Cheers. RobHar (talk) 04:36, 29 October 2008 (UTC)
I don't have the knowledge to make it accessible, but I can write a history up, where should the history section be? You guys should consider me a general reader on this topic, one that knows calculus and a few other unrelated math areas. —— nixeagle 18:58, 29 October 2008 (UTC)
Sounds good. My opinion is that the history section should be pretty much the first section. Certainly, I would place the history section first now. I don't know how other people feel though... Thanks. RobHar (talk) 19:14, 29 October 2008 (UTC)
Ok, I'm going to work on it in /history I have to gather up some book sources, but I think I have some decent internet sources. Our main problem here is that this history of this topic also intertwines with other sub topics of number theory. —— nixeagle 18:29, 31 October 2008 (UTC)

Perhaps commutative algebra should be mentioned at some point in the article as algebraic number theory uses many of its techniques (not that the commutative algebra article is very enlightening...) LkNsngth (talk) 06:38, 20 June 2009 (UTC)

I'm willing to contribute to this article, but can someone give me some hints which parts should be extended/added? Ringspectrum (talk) 21:06, 17 January 2009 (UTC)

Quite a bit. What I've tried to do so far was add some stuff on basic notions so that the rest of the article can be written and make sense (in a more complete version, I would expect much of what I've written to be placed in their own articles). From here, I think it would be useful to start incorporating more recent stuff to better delineate what algebraic number theory is these days such as galois representations, automorphic forms, Iwasawa theory, arithmetic geometry, for example. These each have their own articles (kinda) but some mention of them here would be good. Or maybe it's just that that is what would be easiest for me to add. I think anything that would add content to wiki relevant to algebraic number theory would be good. Do you have any ideas? If so, by all means go ahead. I also have some things I've thought of at User:RobHar/Sandbox3. And of course, the vast and important history of algebraic number theory should be addressed. Cheers. RobHar (talk) 22:15, 17 January 2009 (UTC)
I added a subsection on local fields. You could add things like curves over number fields or Dedekind schmes and Abelian varieties/schemes to your list. —Preceding unsigned comment added by Ringspectrum (talkcontribs) 22:32, 17 January 2009 (UTC)
The history and historical development of the subject is sorely missing from this article. Damien Karras (talk) 12:36, 29 January 2009 (UTC)
I also noticed that while mentioned once above, there is currently no talk about including open problems. Adding some would possibly help give the reader an idea of some of the motivating questions in the field. Also, Fermats last theorem is never mentioned in the article.LkNsngth (talk) 17:42, 20 June 2009 (UTC)
Another comment: somewhere in the article, someone should explain what the notation ${\displaystyle Z[{\sqrt {5}}]}$ means. I would add it but it seems to not fit anywhere LkNsngth (talk) 04:27, 25 June 2009 (UTC)

Perhaps someone can add a section talking about the Langlands program (I know there is another page on it, but it is kind of small). As I understand, this is a important subject in number theory. —Preceding unsigned comment added by LkNsngth (talkcontribs) 04:43, 25 November 2009 (UTC)

## Content

This article is very well written and covers a few important topics. However, in my opinion, algebraic number theory is a topic which can be explained (if done well) to a layman. I understand that one of the implications of this would be that the article is too basic, and sacrifices the formal definitions for "intutive discussions" (this occurs quite frequently with WP articles). Should there be some intent of explaining this concept in a manner which does not require people to have a good grasp of ring theory? I feel that this article should at least appeal to a student of linear algebra and number theory, who has seen some of the general ideas within algebra, and understands the basic questions within number theory. As a side note, I also wonder whether this article will, at some point, no longer be the mathematics collaboration of the month. --PST 08:35, 21 September 2009 (UTC)

Indeed, what is currently here was my attempt to throw a bunch of content onto wiki which was not present before, eventually moving the content to separate new articles. I figured step 1 of writing a nice article on algebraic number theory would be to have the details available via a wikilink. My view for this article would indeed be a much more layman-friendly description of the field of algebraic number theory, relegating the details to subarticles. For example, since my edits, Jakob.scholbach has much improved the algebraic number field article, which is probably a better place for details about algebraic number fields than this article. I think the subarticles should be somewhat specific like Unit group of an algebraic number field. If you'd like to move around the content to make room for a nice big-picture oriented article here, that would be appreciated (by me at least).
As for the collaboration of the month aspect, that project seems to be rather dead. If you'd like to revive it (and change the COTM), I don't think anyone would oppose that. There was some discussion on its talk page about how to make it more popular. Nothing conclusive though. RobHar (talk) 19:07, 21 September 2009 (UTC)

## Orphaned references in Algebraic number theory

I check pages listed in Category:Pages with incorrect ref formatting to try to fix reference errors. One of the things I do is look for content for orphaned references in wikilinked articles. I have found content for some of Algebraic number theory's orphans, the problem is that I found more than one version. I can't determine which (if any) is correct for this article, so I am asking for a sentient editor to look it over and copy the correct ref content into this article.

Reference named "Singh":

Reference named "Elstrodt":

I apologize if any of the above are effectively identical; I am just a simple computer program, so I can't determine whether minor differences are significant or not. AnomieBOT 16:36, 22 October 2013 (UTC)