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.

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    $\begingroup$ 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. $\endgroup$ – paul garrett May 5 '18 at 18:23
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    $\begingroup$ I second what Paul said above, but also of course. My department is chock full of algebraic number theorists $\endgroup$ – leibnewtz May 5 '18 at 18:27
  • $\begingroup$ @paulgarrett Oh, I see. So maybe what I'm after is not really number theory, but something more like commutative algebra, right? $\endgroup$ – rmdmc89 May 5 '18 at 18:28
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    $\begingroup$ @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 $\endgroup$ – leibnewtz May 5 '18 at 18:30
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    $\begingroup$ @lhf, ah, yes, I thought I'd seen such a question somewhere... $\endgroup$ – paul garrett May 5 '18 at 18:51

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.

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