Re: Why is it called a Divisor?

In article <f2giof$qs6$1@xxxxxxxx>, Kira Yamato <kirakun@xxxxxxxxxxxxx> wrote:
For an smooth irreducible algebraic curve C, the divisor is defined to
be a finite formal sum of the form
sum_{i=1...n} z_i p_i
where z_i are integers and p_i are points on C.

I know some basic properties of divisors, like the equivalence class of
divisors mod linear equivalence is isomorphic as a group to the group
of invertible sheaves (mod sheaf-isomorphisms) on C.

But I'm still puzzled why the name 'divisor'?

The following information can be found in Harold M. Edwards "Divisor
Theory", published by Birkhauser.

Divisors of curves are an application of the ideas of Kronecker that
arouse from considerations of algebraic number theory, to the
situation of function fields. The similarities between the function
field case and the number field case were observed by Dedekind and
Weber, who transplanted the theory from the latter to the former
(Dedekind, Richard, and Weber, Heinrich, "Theorie der algebraischen
Funktionen einer Veranderlichen", J. fur Math., 92 (1882),
pp. 181-290).

In their original context of number fields, "divisor" comes from
considerations similar to those of Gauss for quadratic forms;
Kronecker was attempting to define "greatest common divisor" in
situations where you do not have unique factorization (number fields
that are not UFDs).

For a polynomial with integer coefficients, the "content" is defined
to be the greatest common divisor of the coefficients. For polynomials
with rational coefficients, if you factor out denominators, then the
content of a product is the product of the contents (a consequence of
Gauss's Lemma). We want the same to be true for polynomials with
algebraic coefficients, but the notion of "greatest common divisor",
necessary for the definition of content, has no meaning. The theory of
divisors defines the content of a polynomial with algebraic
coefficients in a way that makes the content of a product equal to the
product of the contents.

Let f and g be polynomials in any number of indeterminates, with
algebraic coefficients. There is a polynomial h, with algebraic
coefficients, such that f*h has coefficients in Z. The content of f
divides the content of g if and only if the content of f*h divides the
content of g*h. But since f*h has coefficients in Z, its content is a
positive integer (the greatest common divisor of the coefficients of
f*h, in the usual sense). So "the content of f*h divides the content
of g*h" has a meaning that we already understand.

A "divisor" is the content of a polynomial; in fact, Kronecker never
used the word "content", calling it instead "the divisor represented
by f." The nonzero divisors form a multiplicative group. If you
specify an algebraic number field K and restrict consideration to
polynomials with coefficients in K, then the multiplicative group of
divisors in this sense coincides with the group of ideals in K in the
sense of Dedekind, each "divisor" corresponding to the ideal generated
by the coefficients of the corresponding polynomial f. Thus, these
"divisors" are actually about divisibility, and in particular about
greatest common divisor (which can be defined in terms of ideals).

Dedekind redeveloped Kronecker's theory (for some, like Edwards,
obscuring its vitues), and then applied it to curves (in the form of
function fields), and kept the name of the corresponding concept.

--
======================================================================
"It's not denial. I'm just very selective about
what I accept as reality."
--- Calvin ("Calvin and Hobbes" by Bill Watterson)
======================================================================

Arturo Magidin
magidin-at-member-ams-org

.

Relevant Pages

  • Re: Why is it called a Divisor?
    ... But I'm still puzzled why the name 'divisor'? ... For a polynomial with integer coefficients, ... We want the same to be true for polynomials with ... greatest common divisor (which can be defined in terms of ideals). ...
    (sci.math)
  • Re: g-adic development homework
    ... > a review from someone who knows Lisp. ... (My corrector already told me, ... Instead of increasing the divisor at each step, ... To limit the number of coefficients to N, ...
    (comp.lang.lisp)
  • Re: An observation of Weils
    ... homogeneous polynomials with integer coefficients, ... You need the division algorithm, that is, you need to know ... But any common divisor will, ...
    (sci.math)
  • Re: An observation of Weils
    ... homogeneous polynomials with integer coefficients, ... the greatest common divisor d of two integers p,q ... You need the division algorithm, that is, you need to ...
    (sci.math)
  • Re: How can i find out the quotient?
    ... This is division of polynomials over a field with two elements. ... When it's divisor times x^n, write a '1' someplace; when it's zero, write '0' going ... When you can't divide anymore, ...
    (sci.math)