User talk:Garald/Number theory

Algebraic number theory

I guess one comment I'd like to make is with regards to what algebraic number theory is. I feel pretty strongly that algebraic number theory is the study of algebraic properties of the objects of arithmetic interest by any means necessary, not simply by algebraic methods. Even some fairly old things such as class field theory (which I would place firmly in the "algebraic number theory" box) had several proofs which were initially only known via analytic techniques. In more modern times, for example, modular forms and L-functions have become a major ingredient in much of algebraic number theory (some of the questions surrounding these objects certainly belong to analytic number theory, but others I would surely place in algebraic number theory). There's also all the p-adic analysis (and p-adic functional analysis even) occurring in Iwasawa theory, p-adic Hodge theory, the theory of Galois deformations, and the theory of p-adic families of automorphic forms. I'll leave it at that for now.

A more minor comment, still on the subject of algebraic number theory, regards the paragraph on ideals. I'd like to point you to Franz Lemmermeyer's answer to a mathoverflow question (as well as to a paper he has on the subject available at [1]). Basically, he says that Kummer's theory of ideal numbers grew out of his desire to study higher reciprocity laws rather than any interest in Fermat. While your current phrasing in the last sentence of the paragraph might be true regarding the more widespread study of ideals, I think it's important to also recognize the influence the development of higher reciprocity laws had on the theory of ideals.

I haven't taken a look at the rest of the article yet, but I will shortly. And thank you for your efforts, the number theory article really needs a more expert touch. RobHar (talk) 16:42, 14 August 2010 (UTC)[reply]

Thanks - this is useful. I'll probably make the appropriate edits from Hyderabad next week. Garald (talk) 05:36, 17 August 2010 (UTC)[reply]

Got that done. Garald (talk) 02:07, 8 September 2010 (UTC)[reply]

Comments

I'll add some comments here as I go read along.

Re the staring point of analytic number theory: a while ago, i looked into this for the analytic number theory article and it seemed that Dirichlet's 1837 paper is often taken as the starting point. See the aforementioned wiki article for sources for this statement; namely, Davenport starts his book with this statement, and Apostol also says it in the intro to his undergrad book. RobHar (talk) 23:56, 7 October 2011 (UTC)[reply]

Re your comments on my tak page: Perhaps it would be best to say something like: "There is no agreed upon "beginning" of analytic number theory, but some of the works cited as such include Euler's proof of the infinitude (year and ref to Iwaniec–Kowalski), Dirichlet's theorem on primes in arithmetic progression (1837, ref to Davenport and Apostol), and Riemann's memoir on the Riemann zeta function (year, ref needed!)". How does that sound? I think it's usually best to acknowledge when there's not consensus on something and offer the major possibilities. RobHar (talk) 17:11, 8 October 2011 (UTC)[reply]