Talk:Artin reciprocity

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Mathematics rating: Start Class High Priority  Field: Number theory

are you guys serious or is all this made up?

[edit] Connection to more basic material?

The ignorance of the above comment aside, it would be nice to see how this connects to the more familiar types of reciprocity laws. As I am not a number theorist, I have no idea how involved such an explanation would be. At any rate, this article needs more, so I'll probably tag it when I figure out which tag is the right one to stick on here. VectorPosse 11:02, 3 March 2007 (UTC)

Okay, so I added an expert tag. It's not that I don't believe the material presented is correct; I just think an expert is required to add good material. VectorPosse 11:18, 3 March 2007 (UTC)
Yes, this is not the way to present Artin reciprocity! This "one version of the theorem" may be very useful in Langlands program, but it is neither a standard form of Artin's reciprocity law, nor is it useful for comprehension. I am a bit rusty on my global class field theory, but if no one fixes it, I will (eventually). Arcfrk 11:11, 10 March 2007 (UTC)
To answer VectorPosse's question above, the standard way of showing power reciprocity laws from the Artin reciprocity law requires introducing Hilbert symbols and computing them (this is done in section VIII.3 of Milne's Class Field Theory notes referenced in the article). As for the presentation of the Artin reciprocity law in this article, though not necessarily useful comprehension, this is probably the "cleanest" way to state it, and is definitely an extremely relevant and widely used interpretation of the theorem. I do definitely think a more classical statement should be given as well. RobHar 08:28, 24 March 2007 (UTC)

You mathematicians may scoff at the top comment but it actually makes a serious point. This and too many other mathematical wikipedia articles make absolutely no attempt to explain the topic in a way that is at least half understandable to the masses. Maybe you think this is the kind of topic that that only an expert can begin to comprehend but I think that is a failure of vision on your part. There have been many wonderful books written for general audiences about advanced mathematics (e.g. books on how Fermat's Last Theorem was proven and the search for a proof of the Riemmann Hypothesis) that have not shied away from at least giving a glimpse of some of the techniques involved. The way this page dives straight into highly technical terminlogy and equations presents a virtually infinite barrier to entry for the uninitiated. You guys get a D- in my book--Julian Brown (talk) 22:10, 5 June 2008 (UTC).

As someone who has contributed to this article I would just like to say that the place where you're comment is wrong, Julian, is in assuming that we, the mathematicians, are content with this article. I would love to make this article more understandable, and really say something nice and relevant, but that is unfortunately a very hard thing to do. I have not come across any account of Artin reciprocity that attempts to make its essence understood to the masses (and frankly in some sense it should only be upon finding such a reference that one should add anything to this wikipedia article, since wikipedia is an encyclopedia and should record what is known, not be a primary source). Many of the articles on higher-level mathematics are edited by an extremely small group of people attempting to put the bare bones of the idea in the article, and do not have the kind of community that may allow other, more popular, wikipedia articles to grow in comprehensiveness and comprehensibility. A project such as Wikipedia:Mathematics Collaboration of the Month attempts to create focussed communities to improve targeted math articles, perhaps an attempt to increase interest in such a project would aide in attaining the goal you seek.RobHar (talk) 01:50, 6 June 2008 (UTC)
And beyond RobHar's well-stated point, there is a degree to which advanced articles should be allowed to be advanced. I am a trained mathematician, but I don't pretend that I should be able to understand this article, even when it's well-written and entirely fleshed-out. Class field theory is complicated stuff and people spend years trying to understand even its basics. It's a bit presumptive to come here and pretend like anyone from off the street should be able to read this article once and understand it. I don't think this is an elitist way to think about it. Math is hard sometimes. The most we can hope for is that the lede is sufficiently "blue" that one could work their way back through the links and try to get some context for understanding this necessarily specialized article. VectorPosse (talk) 03:20, 6 June 2008 (UTC)
I don't expect the whole article to be easily accessible. But I think every entry in Wikipedia should start off with at least of couple of sentences (ideally a paragraph or two) that are intelligible to non-experts. I'm a firm believer in the idea that you should gradually walk your readers into deeper waters. This entry makes absolutely no effort to do that and as such is almost useless for someone who is not already steeped in the subject. I appreciate it might be hard to do in this case but I doubt very much that it is not possible. I wish I knew enough about the subject to make an attempt myself but I came here after talking with Ed Witten who said that he was working on the Geometric Langlands Program. I was hoping to get an insight into this subject but every associated entry I have read in Wikipedia is like reading hieroglyphics without a Rosetta Stone. --Julian Brown (talk) 07:41, 6 June 2008 (UTC)
I agree with what you're saying and the intro should definitely (and likely could) be made more accessible. This touches on one of the other points I made above, that there simply aren't very many editors around. In terms of how important Artin reciprocity is as an achievement in number theory, this article is ridiculously small and incomplete, but is actually better than the article on algebraic number theory itself, and if there were more editors around I think it would create more excitement about writing these articles. There are tonnes of articles I want to fix up, but time is not something I have a lot of. I've made some very minor edits to try to make it clearer as to why Artin reciprocity is related to Langlands. Geometric Langlands is a ridiculously complicated thing to set up, the only thing I'd feel comfortable saying on the subject is that in the classial Langlands correspondence one deals (approximately) with Galois representations, and Galois groups are etale fundamental groups and under an analogue of the Riemann-Hilbert correspondence representations of the etale fundamental group correspond to vector bundles with flat connection; as for the automorphic side, there I Hecke eigensheaves and I have no idea what those are. Here's something written by one of the leaders in the field that has some stuff on Langlands and geometric langlands that should have some understandable parts [1]. RobHar (talk) 09:53, 6 June 2008 (UTC)