TITLE: Is there much difference between Kronecker's and Dedekind's methods in algebraic number theory and commutative algebra?
QUESTION [13 upvotes]: Edwards, in his book "Divisor theory" says that Kronecker's methods are quite different to Dedekind's and those of today. Is there really much of a difference apart from Kronecker's methods being more constructive?

REPLY [7 votes]: For a good introduction to the different approaches of Kronecker and Dedekind you should read the relevant sections in Weyl's "Algebraic theory of numbers", Princeton UP 1940. He strongly
favors Kronecker's approach, at least if the goal is saving unique factorization. In fact, Kronecker's methods do what they're supposed to do in the polynomial ring $R = {\mathbb Z}[X]$ (in particular, they show that $R$ is a UFD), whereas Dedekind's ideal theory fails miserably in this respect since $R$ is a UFD yet its ideal theory is complicated as $R$ is not even a Dedekind domain.
Taking up a comment made by Victor Protsak, the divisor theory developed by Borevich and Shafarevich (which I also neglected until very recently) isn't that foreign at all: their "rings with a divisor theory" coincide, unless I am mistaken, with the modern notion of Krull domains. I do not (yet) know any serious and readable account in English (B & S have superfluous axioms; see Gundlach's book below). For those comfortable in German I suggest 

Olaf Neumann, Was sollen und was sind Divisoren? (What are divisors and what are they good for?),  Math. Semesterber. 48, No. 2, 139-192 (2001)
F. Lucius, Ringe mit einer Theorie des grössten gemeinsamen Teilers
(Rings having a theory of greatest common divisor), Diss. Goettingen 1996
K.-B. Gundlach, Einführung in die Zahlentheorie, BI 1972