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Asked differently: Is Algebraic Number Theory the study of the theory of algebraic numbers? Or is it Number Theory from an algebraic viewpoint?

Or is it both?

I know I can just find a wiki article but I figure answers from the MSE community would be more intuitive and instructive.

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    $\begingroup$ en.wikipedia.org/wiki/Algebraic_number_theory $\endgroup$ – Simon S May 20 '15 at 1:04
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    $\begingroup$ Both.$\,\,\,\,$ $\endgroup$ – Zev Chonoles May 20 '15 at 1:04
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    $\begingroup$ @BillDubuque I guess my question can be rephrased as: Is Algebraic Number Theory the study of the theory of algebraic numbers, or is it number theory from an algebraic viewpoint? $\endgroup$ – Al Jebr May 20 '15 at 2:22
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    $\begingroup$ The smart-aleck response is that Algebraic Number Theory is the title of a book by Frazer Jarvis from Springer-Verlag. Or maybe the one by Jürgen Neukirch and Norbert Schappacher, or the one by John William Scott Cassels and Albrecht Frhlich. $\endgroup$ – David R. May 20 '15 at 21:50
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    $\begingroup$ @ZevChonoles In other words, the name satisfies the law of associativity. $\endgroup$ – Jack M May 20 '15 at 22:20
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It's mostly the latter: the study of number theory from an algebraic viewpoint, just as analytic number theory is the study of number theory from the viewpoint of analysis.

With algebraic number theory, it is often easier to solve equations that would be more difficult if not impossible with elementary methods. Algebraic number theory often deals with these equations in the context of a specific (though not necessarily specified) algebraic structure known as a ring, often invoking algebraic concepts like homomorphisms, bijections, surjections, etc.

But of course the distinction between algebraic numbers and algebraic integers is important to know.

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I must disagree with claims that "Algebraic Number Theory" is an algebraic study of anything-whatsoever, possibly including number theory, or, possibly "numbers", whatever the reference may be.

That is, in genuine practice, it is "the theory of algebraic numbers", including "algebraic integers", including $p$-adic methods, including complex variables methods, including harmonic analysis methods, including Galois theory, including rudimentary commutative algebra, ...

E.g., there is (to my knowledge) no "purely algebraic" proof of the analytic continuation and functional equation of zeta functions of number fields, of Hecke L-functions thereof, nor even of Dirichlet's Units Theorem and finiteness of class number... in part because these are not "purely algebraic" facts, because they hold for rings of algebraic integers (and the function field analogues), not for general Dedekind domains.

True, the fact that a little commutative algebra and a little field theory enter might cause some to think that "this is algebra", just as the entrance of some complex analysis induces some to say "it's analytic number theory", but these are essentially irrelevant ways of appraising the situation, and, also, of parsing the names of things.

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  • $\begingroup$ The current description of the algebraic-number-theory tag seems to support your answer more than mine. $\endgroup$ – user153918 May 21 '15 at 15:38
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    $\begingroup$ @AlonsodelArte, I do think that "algebraic number theory" and "analytic number theory" are not optimally distinguished by use of "algebra" versus use of "analysis", despite the words. The supposed "algebra/analysis" schism does not reflect practice, although it is echoed in standard curricula, and, therefore, in textbooks written to fit into that curriculum, etc. $\endgroup$ – paul garrett May 21 '15 at 15:48
  • $\begingroup$ This is interesting. It's contrary to what I thought algebraic number theory was: number theory using techniques and methods from Algebra. Now I see answers arguing for both sides. $\endgroup$ – Al Jebr May 22 '15 at 17:15
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    $\begingroup$ @AlJebr, certainly "algebra" is used in studying algebraic numbers, but not only! Also complex analysis, Fourier analysis, and many other things. $\endgroup$ – paul garrett May 22 '15 at 17:19
  • $\begingroup$ @paulgarrett Algebra as in algebraic structures: groups, rings, fields? $\endgroup$ – Al Jebr May 22 '15 at 17:23
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Consider the Wikipedia page for Algebraic Number Theory in other languages:

The exception that proves the rule is Spanish:

which starts by acknowledging the other form: "La teoría de números algebraicos o teoría algebraica de números ..."

In these languages (which are the ones I can make some sense of), it is clear that the theory is algebraic, not the numbers. On the other hand, it does study algebraic numbers, hence the confusion.

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    $\begingroup$ It would be more productive to review the math.SE tags number-theory, elementary-number-theory. algebraic-number-theory and analytic-number-theory. Presumably there is a lot more authorial accountability editing the math.SE tag wiki than there is editing Wikipedia. $\endgroup$ – user153918 May 21 '15 at 15:37
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It is the study of number theory from an algebraic viewpoint. The methods of algebraic number theory are used to solve many problems in number theory. For example, the study of Gaussian integers sheds light on problem of which prime numbers are the sum of two squares.

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  • $\begingroup$ Not sure why, but your answer as posted is incomplete. I understand what you were getting at bringing up Gaussian integers and prime numbers as sums of two squares. But beyond that, I'd have to read your mind. $\endgroup$ – Robert Soupe May 21 '15 at 2:34
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Not only are the answers from the Mathematics StackExchange community "more intuitive and instructive," they're much more valid than anything you will find on Wikipedia. Although it's true that a lot of the "community" here are also active on Wikipedia, their talents and insights are mostly wasted over there.

There is much tighter control here than there. And not just anyone with an account here can edit the tag list (for example, I can't, however much the lower-case "diophantine" may bother me). That tag list defines algebraic-number-theory thus:

Questions related to the algebraic structure of algebraic integers

Seems very clear to me. For the sake of comparison, look at elementary-number-theory

Questions on congruences, linear diophantine equations, greatest common divisor, divisibility, etc.

analytic-number-theory

Questions on the use of the methods of real/complex analysis in the study of number theory.

and p-adic-number-theory

In mathematics the p-adic number system for any prime number p extends the ordinary arithmetic of the rational

I would adjust the punctuation of that last one, but, like I said, I can't edit the tag descriptions. But I can go on Wikipedia right now and insert all manners of nonsense and wrongheadedness.

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