Is 'Algebraic Number Theory' the study of the theory of algebraic numbers, or is it the study of the theory of numbers from an algebraic viewpoint?

Question

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.

Answer 1

Consider the Wikipedia page for Algebraic Number Theory in other languages:

• Théorie algébrique des nombres

• Algebraische Zahlentheorie

• Teoria algebrica dei numeri

• Teoria algébrica dos números

The exception that proves the rule is Spanish:

• Teoría de números algebraicos

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.

Answer 2

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.

Answer 3

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.

Answer 4

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.

Answer 5

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.