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I was wondering if number theorists are "number theorists," or eventually resolve themselves into one of the various branches - i.e., algebraic, analytic, etc.

Also out of curiosity, I was wondering if there is a particular branch or area that draws the most attention today.

(Full disclosure: this is just of interest. I am by no means in a position to utilize such an answer, but in that I am pretty much in math isolation, this is my only forum to find out anything.)


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Dear Andrew, This answer describing some of the different areas of number theory might be of interest. Also, Ireland and Rosen is a great book! (I routinely recommend it to my students.) Regards, – Matt E Jun 28 '12 at 4:32
@MattEDear Matt, Thanks for taking the time to point me towards that answer. (I'm a bit embarrassed in that I should have searched for it myself.) I actually just got a copy of Ireland and Rosen - great, as you say. In fact I was thinking about adding some remark to this question I had asked.… Of course Dylan's always encouraging comments are appreciated. But I hit the wall in Samuel at the end of Ch. 2 with discriminants - am regrouping with I & R. Just happy to be in the ANT game. – TheBirdistheWord Jun 28 '12 at 22:47
up vote 7 down vote accepted

Yes, different areas of number theory can be very, very different. At my university, I'm surrounded by many different number theorists, and half of us can barely understand the other half. I consider myself a rising analytic number theorist and I'll be working close to 4 other analytic number theorists (Jeff Hoffstein's 4 students and me, a hopeful 5th). And between the five of us, we don't come close to understanding a single thing done by the other major corps of number theorists here (Joe Silverman's 4 students).

To skimp on lots of details, we do analytic number theory and they do algebraic number theory. And this is all different from the number theory I looked into during my undergrad at Georgia Tech. I would now call that Additive Combinatorics or Additive Number Theory, but I think of most of the people who research in that area as analytic number theorists (just different than my cup of analytic-number-theory-tea).

And somehow, there are many disparate parts of number theory. I would say that the three areas I mentioned are also pretty big in drawing lots of attention: the Langlands Program is pretty big right now and has a lot to do with automorphic forms. Elliptic Curves are also pretty big. And Terry Tao does a lot with arithmetic and additive number theory, and thus a lot tends to get done.

But I should mention, to not give you the completely wrong idea, that there is some interplay between the different branches of number theory. I like to think that I'm going to do a very algebraic form of analytic number theory, and questions related to elliptic curves might even come up now and again. But number theory is a large field, all the larger because it tends to be classified by content rather than method (as opposed to group theory, analysis, harmonic analysis, ode and pde, etc.).

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@mixedmathNice answer. How ironic; what prompted me to ask this question was I have just ordered Ireland and Rosen. Clearly there is a substantial algebraic content in the early chapters. But then there is a later chapter entitled Algebraic Number Theory, so I was wondering if that topic was confined to that chapter - but thanks to your answer, I see ANT pervades the book. – TheBirdistheWord Jun 22 '12 at 0:32
@mixedmathI should have said "How ironic - Brown" – TheBirdistheWord Jun 22 '12 at 0:57
Dear mixedmath, Thanks again for this answer. I started to look into analytic number theory; I had previously focused on the algebraic side. I started with Stopple's book. Quite nice and gentle. I somewhat would like to pick up the pace, and am inclined to go with Apostol. I have seen some reference requests in this regard. Yet I would very much appreciate your personal opinion as to this choice or if you recommend a different text or other source (i.e., notes, etc.). I hope this is not an imposition. Thanks and best regards, – TheBirdistheWord May 25 '13 at 12:13
P.S. Since I am a self-studier and don't have to adhere to a curriculum, I am quite open to anything that's beautifully written. Thanks again. – TheBirdistheWord May 26 '13 at 20:48
@Andrew I really like Apostol's book, and I give it a high recommendation. – mixedmath May 27 '13 at 4:33

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