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?

Solution 1:

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.

Solution 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.

Solution 3:

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.