Dedekind Domain - The Prehistory of Dedekind Domains

The Prehistory of Dedekind Domains

In the 19th century it became a common technique to gain insight into integral solutions of polynomial equations (i.e., Diophantine equations) using rings of algebraic numbers of higher degree. For instance, fix a positive integer . In the attempt to determine which integers are represented by the quadratic form, it is natural to factor the quadratic form into, the factorization taking place in the ring of integers of the quadratic field . Similarly, for a positive integer the polynomial (which is relevant for solving the Fermat equation ) can be factored over the ring, where is a primitive root of unity.

For a few small values of and these rings of algebraic integers are PIDs, and this can be seen as an explanation of the classical successes of Fermat and Euler . By this time a procedure for determining whether the ring of all algebraic integers of a given quadratic field is a PID was well known to the quadratic form theorists. Especially, Gauss had looked at the case of imaginary quadratic fields: he found exactly nine values of for which the ring of integers is a PID and conjectured that there are no further values. (Gauss' conjecture was proven more than one hundred years later by Heegner, Baker and Stark.) However, this was understood (only) in the language of equivalence classes of quadratic forms, so that in particular the analogy between quadratic forms and the Fermat equation seems not to have been perceived. In 1847 Gabriel Lamé announced a solution of Fermat's Last Theorem for all -- i.e., that the Fermat equation has no solutions in nonzero integers, but it turned out that his solution hinged on the assumption that the cyclotomic ring is a UFD. Ernst Kummer had shown three years before that this was not the case already for (the full, finite list of values for which is a UFD is now known). At the same time, Kummer developed powerful new methods to prove Fermat's Last Theorem at least for a large class of prime exponents using what we now recognize as the fact that the ring is a Dedekind domain. In fact Kummer worked not with ideals but with "ideal numbers", and the modern definition of an ideal was given by Dedekind.

By the 20th century, algebraists and number theorists had come to realize that the condition of being a PID is rather delicate, whereas the condition of being a Dedekind domain is quite robust. For instance the ring of ordinary integers is a PID, but as seen above the ring of algebraic integers in a number field need not be a PID. In fact, although Gauss also conjectured that there are infinitely many primes such that the ring of integers of is a PID, to this day we do not even know whether there are infinitely many number fields (of arbitrary degree) such that is a PID! On the other hand, the ring of integers in a number field is always a Dedekind domain.

Another illustration of the delicate/robust dichotomy is the fact that being a Dedekind domain is, among Noetherian domains, a local property -- a Noetherian domain is Dedekind iff for every maximal ideal of the localization is a Dedekind ring. But a local domain is a Dedekind ring iff it is a PID iff it is a discrete valuation ring (DVR), so the same local characterization cannot hold for PIDs: rather, one may say that the concept of a Dedekind ring is the globalization of that of a DVR.