Many more authors state theorems for Dedekind domains with the implicit proviso that they may require trivial modifications for the case of fields.
In the 19th century it became a common technique to gain insight into integer solutions of polynomial equations using rings of algebraic numbers of higher degree.
, the factorization taking place in the ring of integers of the quadratic field
these rings of algebraic integers are PIDs, and this can be seen as an explanation of the classical successes of Fermat (
By this time a procedure for determining whether the ring of all algebraic integers of a given quadratic field
Especially, Gauss had looked at the case of imaginary quadratic fields: he found exactly nine values of
(Gauss's conjecture was proven more than one hundred years later by Kurt Heegner, Alan Baker and Harold Stark.)
At the same time, Kummer developed powerful new methods to prove Fermat's Last Theorem at least for a large class of prime exponents
On the other hand, the ring of integers in a number field is always a Dedekind domain.
In fact, this is the definition of a Dedekind domain used in Bourbaki's "Commutative algebra".
of algebraic integers in a number field K is Noetherian, integrally closed, and of dimension one: to see the last property, observe that for any nonzero prime ideal I of R, R/I is a finite set, and recall that a finite integral domain is a field; so by (DD4) R is a Dedekind domain.
The other class of Dedekind rings that is arguably of equal importance comes from geometry: let C be a nonsingular geometrically integral affine algebraic curve over a field k. Then the coordinate ring k[C] of regular functions on C is a Dedekind domain.
This is largely clear simply from translating geometric terms into algebra: the coordinate ring of any affine variety is, by definition, a finitely generated k-algebra, hence Noetherian; moreover curve means dimension one and nonsingular implies (and, in dimension one, is equivalent to) normal, which by definition means integrally closed.
Both of these constructions can be viewed as special cases of the following basic result: Theorem: Let R be a Dedekind domain with fraction field K. Let L be a finite degree field extension of K and denote by S the integral closure of R in L. Then S is itself a Dedekind domain.
Taking R = Z, this construction says precisely that rings of integers of number fields are Dedekind domains.
Zariski and Samuel were sufficiently taken with this construction to ask whether every Dedekind domain arises from it; that is, by starting with a PID and taking the integral closure in a finite degree field extension.
[6] If the situation is as above but the extension L of K is algebraic of infinite degree, then it is still possible for the integral closure S of R in L to be a Dedekind domain, but it is not guaranteed.
Given two fractional ideals I and J, one defines their product IJ as the set of all finite sums
for some nonzero x in K. Note that each principal fractional ideal is invertible, the inverse of
is a unit in R. For a general domain R, it is meaningful to take the quotient of the monoid Frac(R) of all fractional ideals by the submonoid Prin(R) of principal fractional ideals.
We note that for an arbitrary domain one may define the Picard group Pic(R) as the group of invertible fractional ideals Inv(R) modulo the subgroup of principal fractional ideals.
A remarkable theorem of L. Claborn (Claborn 1966) asserts that for any abelian group G whatsoever, there exists a Dedekind domain R whose ideal class group is isomorphic to G. Later, C.R.
In 1976, M. Rosen showed how to realize any countable abelian group as the class group of a Dedekind domain that is a subring of the rational function field of an elliptic curve, and conjectured that such an "elliptic" construction should be possible for a general abelian group (Rosen 1976).
In contrast, one of the basic theorems in algebraic number theory asserts that the class group of the ring of integers of a number field is finite; its cardinality is called the class number and it is an important and rather mysterious invariant, notwithstanding the hard work of many leading mathematicians from Gauss to the present day.
In view of the well known and exceedingly useful structure theorem for finitely generated modules over a principal ideal domain (PID), it is natural to ask for a corresponding theory for finitely generated modules over a Dedekind domain.
Let us briefly recall the structure theory in the case of a finitely generated module
can be decomposed into a direct sum of cyclic torsion modules, each of the form
Remarkably, the additional structure in torsionfree finitely generated modules over an arbitrary Dedekind domain is precisely controlled by the class group, as we now explain.
is the Grothendieck group of the commutative monoid of finitely generated projective
An additional consequence of this structure, which is not implicit in the preceding theorem, is that if the two projective modules over a Dedekind domain have the same class in the Grothendieck group, then they are in fact abstractly isomorphic.