Editing Dedekind domain (section)

== The prehistory of Dedekind domains ==
In the 19th century it became a common technique to gain insight into [[Diophantine equations|integer solutions]] of [[polynomial equation]]s using [[ring (mathematics)|rings]] of [[algebraic number]]s of higher degree.  For instance, fix a positive [[integer]] <math>m</math>.  In the attempt to determine which integers are represented by the [[quadratic form]] <math>x^2+my^2</math>, it is natural to factor the quadratic form into <math>(x+\sqrt{-m}y)(x-\sqrt{-m}y)</math>, the factorization taking place in the [[ring of integers]] of the [[quadratic field]] <math>\mathbb{Q}(\sqrt{-m})</math>.  Similarly, for a positive integer <math>n</math> the [[polynomial]] <math>z^n-y^n</math> (which is relevant for solving the Fermat equation <math>x^n+y^n = z^n</math>) can be factored over the ring <math>\mathbb{Z}[\zeta_n]</math>, where <math>\zeta_n</math> is a [[primitive root of unity|primitive ''n''-th root of unity]].

For a few small values of <math>m</math> and <math>n</math> these rings of algebraic integers are PIDs, and this can be seen as an explanation of the classical successes of [[Pierre de Fermat|Fermat]] (<math>m = 1, n = 4</math>) and [[Leonhard Euler|Euler]] (<math>m = 2,3, n = 3</math>).  By this time a procedure for determining whether the ring of all [[quadratic integer|algebraic integers]] of a given quadratic field <math>\mathbb{Q}(\sqrt{D})</math> is a PID was well known to the quadratic form theorists.  Especially, [[Carl Friedrich Gauss|Gauss]] had looked at the case of imaginary quadratic fields: he found exactly [[Heegner number|nine values]] of <math>D < 0</math> for which the ring of integers is a PID and conjectured that there were no further values.  (Gauss's conjecture was proven more than one hundred years later by [[Kurt Heegner]], [[Alan Baker (mathematician)|Alan Baker]] and [[Harold Stark]].)  However, this was understood (only) in the language of [[equivalence class]]es 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 <math>n > 2</math>; that is, that the Fermat equation has no solutions in nonzero integers, but it turned out that his solution hinged on the assumption that the [[Cyclotomic field|cyclotomic]] ring <math>\mathbb{Z}[\zeta_n]</math> is a UFD.  [[Ernst Kummer]] had shown three years before that this was not the case already for <math>n = 23</math> (the full, finite list of values for which <math>\mathbb{Z}[\zeta_n]</math> 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 number|prime]] exponents <math>n</math> using what we now recognize as the fact that the ring <math>\mathbb{Z}[\zeta_n]</math> is a Dedekind domain.  In fact Kummer worked not with ideals but with "[[ideal number]]s", 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 <math>\mathcal{O}_K</math> of algebraic integers in a [[number field]] <math>K</math> need not be a PID.  In fact, although Gauss also conjectured that there are infinitely many primes <math>p</math> such that the ring of integers of <math>\mathbb{Q}(\sqrt{p})</math> is a PID, it is not yet known whether there are infinitely many number fields <math>K</math> (of arbitrary degree) such that <math>\mathcal{O}_K</math> 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 domain]]s, a [[local property#Properties of commutative rings|'''local''' property]]: a Noetherian domain <math>R</math> is Dedekind iff for every [[maximal ideal]] <math>M</math> of <math>R</math> the [[localization of a ring|localization]] <math>R_M</math> is a Dedekind ring.  But a [[local ring|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.