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?



Is there much difference between Kronecker's and Dedekind's methods in algebraic number theory and commutative algebra?
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). It is at JSTOR [here](https://www.jstor.org/stable/1993332?seq=1#metadata_info_tab_contents). 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.


