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?


You can find a nice description of Kronecker's approach in an article by Harley Flanders, "The Meaning of the Form Calculus in Classical Ideal Theory" (Trans. AMS 95 (1960), 92--100). I found that more to my tastes than Edwards' book.

There is a difference between the two approaches. Kronecker was thinking in very general terms, beyond the "one-dimensional" setting that Dedekind worked in. (Kronecker had a dream -- a second one I suppose -- of unifying number theory and algebraic geometry but the tools to achieve this would take a couple more generations to appear). That accounts in part for Kronecker's multivariable polynomials. He had bigger goals than just unique factorization in rings of integers.

Here is one example of the difference between Kronecker and Dedekind. Suppose ${\mathfrak a}$ is an ideal in the ring of integers of a number field $K$ and I ask you to compute its norm, i.e., the size of ${\cal O}_K/\mathfrak a$. How would you do it? From Dedekind's point of view, you find a ${\mathbf Z}$-basis of ${\cal O}_K$ and of ${\mathfrak a}$, write the basis of the ideal in terms of the basis of the ring of integers, and then compute (the absolute value of) the determinant of the matrix expressing the ideal basis in terms of the ring basis. But as you may know, ideals usually are not given to us in terms of a ${\mathbf Z}$-basis. More often they are given in terms of just two generators, say ${\mathfrak a} = (\alpha,\beta)$. How can you compute the norm of the ideal in terms of the two generators? In principle it should be possible, since the two generators determine the ideal they generate, so all the data you need is encoded in the numbers $\alpha$ and $\beta$.

There is a Dedekind-style way to write the norm of ${\mathfrak a}$ in terms of the two generators: the norm of an ideal is the gcd of the norms of all elements of the ideal. Watch out: you can't get by using only the gcd of the norms of the two generators. For example, in the Gaussian integers the ideal $(1+2i,1-2i)$ is the unit ideal $(1)$, so it has norm 1, but the two generators $1+2i$ and $1-2i$ have norm 5, whose gcd is not 1. (Of course the ideal also contains $1+2i - (1-2i) = 4i$, whose norm is 4, and the gcd of that with 5 is one and you're done.) In principle you only need to form the gcd of the norms of a finite number of elements in the ideal, but it's not clear which "finitely many" elements are practically enough. So I think it's fair to say Dedekind's point of view does not easily allow you to find the norm of an ideal in terms of two generators of the ideal, which is how one usually thinks about them concretely.

Now here is how Kronecker would find the norm of the ideal (essentially). Form the polynomial $\alpha + \beta{T}$ in ${\cal O}_K[T]$. The field extension $K(T)/{\mathbf Q}(T)$ is finite. Take the field norm of $\alpha + \beta{T}$ down to ${\mathbf Q}(T)$. The result is in ${\mathbf Z}[T]$. That integral polynomial has finitely many coefficients (which are not all norms of elements in $K$, so this isn't some disguised version of the previous paragraph). The gcd of the integral coefficients of ${\rm N}_{K(T)/{\mathbf Q}(T)}(\alpha + \beta{T})$ is the norm of the ideal. And if the ideal is given to you with more than two generators, just let $f(T)$ be the polynomial with higher degree having those generators as its coefficients, one for each power of $T$ (it doesn't matter what order you use the generators as coefficients) and do the same thing as in the case of two generators: field norm down to ${\mathbf Q}(T)$ and then gcd of the integral coefficients that pop out. I personally was blown away when I saw this method work, since practically no books on algebraic number theory discuss Kronecker's point of view, so this particular result isn't there. (To be honest, you do not need Kronecker's multivariable polynomial method to prove this norm formula. Once you know the formula, it can be derived by more orthodox techniques, but of course it leaves out the question of how anyone would have ever discovered this formula in the first place by orthodox methods. Any suggestions?)

In a sense this example is only a "constructive" dichotomy between Kronecker and Dedekind, but I think it still addresses the question that is asked, because each method of solving this problem (Dedekind's ${\mathbf Z}$-bases and Kronecker's polynomials) is constructive but they feel so different from each other.

  • $\begingroup$ Are you sure about "second dream"? Hilbert's 12th problem, aleph0.clarku.edu/~djoyce/hilbert/problems.html#prob12, mentions both abelian extensions of imaginary quadratic fields and the analogy between number fields and function fields as parts of a grandiose theory that unifies "three fundamental branches of mathematics, number theory, algebra and function theory" (1st sentence of the last paragraph). My interpretation of it is that this broad analogy was the true Kronecker's dream, but perhaps I am retroactively ascribing Hilbert's point of view to Kronecker himself? $\endgroup$ – Victor Protsak Jun 1 '10 at 4:09
  • $\begingroup$ I was alluding only to Hilbert's 12th problem as the "first dream" and by "second dream" I meant the idea of a general theory encompassing both number fields and function fields (in several variables, as Kronecker may have put it.) I was not sure if these were really part of the same dream of Kronecker's youth. Probably someone could check Vladut's book on modular functions to check. Or maybe some encyclopedic website... $\endgroup$ – KConrad Jun 1 '10 at 4:51

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

  • $\begingroup$ Yes, the "rings with a theory of divisors" in Borevich and Shafarevich are precisely Krull rings if you allow fields to have a theory of divisors with an empty set of discrete valuations (or not, if you don't consider fields to be Krull rings). An English language reference that I used to convince myself the B&S's rings are the same as Krull rings is Matsumura's Commutative Ring theory. In the list of 3 axioms for rings with a theory of divisors in B&S, on p. 171, axioms 1 and 3 imply axiom 2. My reference on that is L. Skula, "Divisorentheorie einer Halbgruppe" Math Z. 114 (1970), 113--120. $\endgroup$ – KConrad Jun 1 '10 at 19:09
  • $\begingroup$ @KConrad An English summary of Lucius' thesis is Rings with a theory of greatest common divisors, Manuscripta Math. 95, 117-136 (1998). This is the best English introduction as far as I know. $\endgroup$ – Bill Dubuque Jan 3 '12 at 6:52

If an opinion counts as an answer, here's mine. In short, Kronecker is pre-modern and Dedekind is modern following more ore less Jeremy Gay's terminology from Plato's ghost: the modernist transformation of mathematics, see also Quinn's paper-in-progress The nature of Contemporary Core Mathematics.

Let me explain. Kronecker was pre-modern in the non-formal use of the concept of divisor. It is used for the same purpose as an ideal: a kind of (substitute of) a common divisor of several elements on a ring. You are not really interested on what a divisor is but on what you can do with it. The trouble is the difficulty in checking the accuracy of your arguments. In any case, everything you do is constructive, so it is tied to specific examples of rings where the constructions are feasible.

The opposite point of view is giving a precise definition in terms of sets. An ideal behaves like a common divisor but has a precise meaning in whatever ring you may think of, i.e. an abstract structure. All methods, constructive or not, are allowed. If we believe Gray (and several other historians) by 1930 pre-modern methods were not used anymore.

It may be worthwhile to remark that Kronecker's divisors were resurrected later by A. Weil in the context of algebraic geometry, but now as a complete formalized thing. Recall that Weil was one of the founders of Bourbaki who tried to put in correct modern way the basics of mathematics.

  • 1
    $\begingroup$ Gray's "Algebraic Geometry Between Noether and Noether" is another good reference. $\endgroup$ – Jeremy Shipley May 31 '10 at 11:55
  • $\begingroup$ "Number theory" by Borevich and Shafarevich develops a notion of a "(commutative) ring with divisor theory". I confess that it had felt so obscure that I never made much headway with it, which I now regret. $\endgroup$ – Victor Protsak Jun 1 '10 at 3:53
  • $\begingroup$ I suppose it's a typo, but I am quite taken with the phrase, "paper-in-pregress," from the 1st paragraph of this answer. $\endgroup$ – Gerry Myerson Jun 1 '10 at 4:01

Urgh - Hilbert stole Kronecker's key ideas on "module theory" to use in invariant theory, while excluding them from his Zahlbericht, and trying to wipe out the "constructive" point of view? Dedekind was more influenced by the analogy with algebraic curves, which Hilbert internalised in conjecturing what would become class field theory? There is a real difficulty of anachronism in trying to decide retrospectively what people were doing in research work, to fit it into later ideas of exposition. Particularly since "modern algebra", which became "abstract algebra", was put together to such an extent to be able to say cleanly what had been discovered in this area.

FWIW, I think Edwards is right as a historian. Kronecker probably operated much more freely in general quotients of rings of integral polynomials. What would proving him wrong look like?


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