This is a somewhat subjective question, about the past, present and especially future of algebraic number theory. I'm not at all in this area, but I'd be interested in an answer.


As we all know, algebraic number theory is one of the oldest and most developed areas in math. New and spectacular results come very often, one can mention here the work of Wiles, or of several recent Fields medalists.


My point now: I might be wrong here, not being a specialist, but my impression is that all this new and spectacular work, and for some time already, rather consists in **building** new things, and not much in substantially **simplifying** the theory.


**Question.** *Is there any big simplification in sight, in algebraic number theory? If so, in which precise direction?*


I'm asking this question because I'm a bit puzzled by what is going on, when compared to other branches of math. Any healthy theory must alternate "building" and "simplifying" periods. By "healthy" here I mean something really viable, that cannot just die over the time from the lack of new and interesting results, but nor from over-complication either.


[**Edited, Feb, 9, 2013**, by A. Caicedo. Original question by [Teo B](https://mathoverflow.net/users/29333/).] 



Algebraic number theory: building and simplifying
A major simplification in algebraic number theory occurred in the beginning of the 20th century when Hensel explicitly introduced his $\mathfrak{p}$-adic numbers. Compare the original cumbersome definition of the Hilbert symbol with the modern definition using local fields.


Another major simplification occurred in the 40s when Chevalley put all these local fields together into the ring of adèles. Compare the original cumbersome formulation of class field theory in terms of ray class groups with the modern formulation in terms of idèle class characters.


A third major simplification occurred in the 50s when Tate rederived results of Hecke and others about the analytic properties (analytic continuation and functional equations) of certain $\zeta$- and $L$-functions using Fourier analysis on the said adèles and idèles.


As far as Wiles's proof of Fermat's Last Theorem is concerned, I believe that many of the ingredients (such as modularity lifting theorems) have since been vastly simplified and generalised. You should ask the experts, or a more specific question here.


So the trend towards greater simplicity and generality will continue, but there is no reason to believe that the proof will one day fit into a small volume accessible to undergraduates. Is there such a proof of the Kronecker-Weber theorem, which is now more than a hundred years old ?


**Addendum** I didn't want to create the impression that simplifications have not been made in recent times. Consider for example the local-to-global principle for the existence of rational points. Various people (Lind, Reichardt, Selmer, Cassels, Swinnerton-Dyer, ...) found examples of the failure of this principle, but these examples could be understood from a unified and simplified perspective only after Manin (1970) introduced his obstruction based on the Brauer group. 


Since then, Skorobogatov (1999) has found examples of the failure of the local-to-global principle which cannot be accounted for by the Manin obstruction. I get the impression that such examples are beginning to be understood from a general point of view only now, and that the theory of étale homotopy is being used to find a whole hierarchy of obstructions; see for example *Homotopy Obstructions to Rational Points* by Yonatan Harpaz and Tomer M. Schlank (<http://arxiv.org/abs/1110.0164>).


It might be objected that the introduction of such a high-level theory into what was initially an elementary question cannot be called a simplification. However, most mathematicians would concur that it is indeed a simplification when a general theory serves to illustrate various disparate phenomena. This simplification has not yet been fully worked out.


