18

I've took an introductory course on Algebraic Number Theory during my Master's, which I really enjoyed. Now that I'm beginning to think about my PhD area, I wonder if I shouldn't go for something in that direction.

However, as in practically all courses I've taken, I was too busy trying to understand definitions and theorems, having little time left (or not enough knowledge) to understand the historical development or the research perspectives on the subject.

Is Algebraic Number Theory still an active area of research? I ask this because almost everything I read or hear about number theory from recent years seems to involve analytic methods (harmonic analysis, probability, ergodic theory etc.), which are not really my cup of tea.

Since I'm a more algebra-driven guy (meaning I'm instinctly attracted to things like Commutative Algebra, Algebraic Geometry, Galois Theory and, of course, Algebraic Number Theory), I wish there were still active lines of research in Number Theory using actual algebraic methods, at least for the most part.

I know this question is rather vague, but that's how well I'm able to articulate it right now, so any advice, insights, reading suggestions etc. would be greatly appreciated.

rmdmc89
  • 10,709
  • 3
  • 35
  • 94
  • 8
    One point is that the name of the subject is chronically misleading: it is really "the theory of algebraic numbers", not "algebraic theory of numbers". That is, the basic results do need some analytical features, from complex analysis and some form of harmonic analysis. The applicability of those ideas is what distinguishes rings of algebraic integers from generic commutative rings, even from generic Dedekind domains, etc., for which the usual number theory results will not hold. – paul garrett May 05 '18 at 18:23
  • 1
    I second what Paul said above, but also of course. My department is chock full of algebraic number theorists – Exit path May 05 '18 at 18:27
  • @paulgarrett Oh, I see. So maybe what I'm after is not really number theory, but something more like commutative algebra, right? – rmdmc89 May 05 '18 at 18:28
  • 2
    @Stephen I think it's okay to want to work in a field where certain methods are used. For example I find many questions in differential geometry fascinating but I prefer the style of argumentation in algebraic geometry – Exit path May 05 '18 at 18:30
  • @paulgarrett, see also https://math.stackexchange.com/questions/1290443/is-algebraic-number-theory-the-study-of-the-theory-of-algebraic-numbers-or-is. – lhf May 05 '18 at 18:40
  • @Stephen but I should have some selectivity, right? I mean, there are many problems out there and as far as my (short) experience goes, if you ask questions which sound purely algebraic, usually the answer is mostly algebraic (using an extreme example, I can't imagine asking about algebraic line bundles and hearing an answer involving harmonic analysis). I guess I'm looking for problems that will most likely have algebraic answers. – rmdmc89 May 05 '18 at 18:46
  • @rmdmc89, yes, I think a better, more descriptive label for what you're wanting is "commutative algebra", in its contemporary sense. – paul garrett May 05 '18 at 18:47
  • 1
    @lhf, ah, yes, I thought I'd seen such a question somewhere... – paul garrett May 05 '18 at 18:51
  • 1
    @rmdmc89 I too find myself in the EXACT same shoes as you and I enjoy more of the algebraic topics, as you mentioned. I don't mind some analytic method but as you mentioned it isn't my cup of tea. It appears that you have decided to go with Algebraic Number Theory/Arithmetic Geometry based on your recent posts. If you don't mind could you share your current experience with the same and whether that really suited your algebraic preference? Could you also mention your reason for going with Number Theory side as opposed to the "pure" Commutative Algebraic side? Thanks in advance! – user600016 Nov 30 '21 at 06:25
  • 1
    @user600016, I wrote this question in the beggining of my PhD, which is almost over, so I do have a couple things to say. To begin with, I'm greateful for my advisor who insisted that I should be familiar with differential geometry, which sounded a little arbitrary to me at the time. She was right. Many crucial ideas in Algebraic Geometry come from differential geometry and complex analysis, and it's just silly to try to understand one thing without the other (I think people should meet Riemann Surfaces, for example, before diving into Algebraic Geometry). – rmdmc89 Dec 17 '21 at 00:14
  • 1
    It's funny that I didn't mention geometry in the post, because it's something I've always liked and involves a big part of what I do now. Of course I was influenced by my advisor, who works in arithmetic geometry, more specifically with elliptic surfaces over number fields (roughly speaking, surfaces made of elliptic curves). What fascinates me about this subject is that, even though it looks very number-theorical (and, by the way, it is), we make heavy use of algebraic geometry. The formal arguments are algebraic, but it's impressive how many times I catch myself thinking visually. – rmdmc89 Dec 17 '21 at 00:15
  • 1
    Frankly I must say that Commutative Algebra, as a purely abstract thing, doesn't attract me the way it used to. Now I can only look at it with the lenses of Algebraic Geometry or Number Theory, otherwise it puts me off. Generally speaking, I think Algebraic Geometry never disappoints me, because it appeals to both to my algebraic and visual sides. It can be intimidating, but a great part of it is due to inadequate teaching or writing. The machinery is heavy, it's true, but the main ideas are very natural once you grab them in your own hands. – rmdmc89 Dec 17 '21 at 00:19
  • 1
    @rmdmc89 Wow thanks a lot for the detailed respond, really appreciate it! – user600016 Dec 17 '21 at 03:04

1 Answers1

29

Nowadays, the distinction between algebraic and analytic number theory is not in the proofs, but in the questions you are trying to answer. Analytic number theory asks questions like "how are the primes distributed on the number line?" Algebraic number theory asks questions like "how do primes split in a given extension of number fields?"

Many questions in algebraic number theory are hard to answer just by using algebra. There has been an enormous amount of insight gained by bringing in analytic techniques. For example, there is no known proof just using algebra that there are no nontrivial unramified extensions of $\mathbb Q$. But the Minkowski bound for the discriminant shows that it is never trivial.

There are some deep questions about integer solutions to polynomial equations which have only been answered by connecting them to modular forms. More generally, the representations and associated L-functions of reductive groups are expected to yield considerable arithmetic insight once the Langlangs Program is complete. This seems to be the most promising direction for the future of algebraic number theory.

If you love algebraic number theory, I would recommend embracing the analytic techniques with the algebraic ones. When you're really doing this kind of math, you won't be able to distinguish whether you are doing algebra or analysis. If you truly dislike analysis, you might be better off doing something like commutative algebra.

D_S
  • 35,843
  • unfortunately I believe most active research areas require both algebraic and analytic techniques, for example even representation theorists need to learn automorphic forms from complex analysis – Hins Jan 14 '25 at 14:28
  • @Hins: What about that state of affairs is unfortunate? – Lee Mosher Jan 20 '25 at 21:49