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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.)

Thanks

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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. math.stackexchange.com/questions/153013/… 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. –  Andrew Jun 28 '12 at 22:47

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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. –  Andrew Jun 22 '12 at 0:32
    
@mixedmathI should have said "How ironic - Brown" –  Andrew 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, –  Andrew 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. –  Andrew 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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