Dedekind domain

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In abstract algebra, a Dedekind domain or Dedekind ring, named after Richard Dedekind, is an integral domain in which every nonzero proper ideal factors into a product of prime ideals. It can be shown that such a factorization is then necessarily unique up to the order of the factors. There are at least three other characterizations of Dedekind domains which are sometimes taken as the definition: see below.

Note that a field is a commutative ring in which there are no nontrivial proper ideals, so that any field is a Dedekind domain, however in a rather vacuous way. Some authors add the requirement that a Dedekind domain not be a field. Many more authors state theorems for Dedekind domains with the implicit proviso that they may require trivial modifications for the case of fields.

An immediate consequence of the definition is that every principal ideal domain (PID) is a Dedekind domain. In fact a Dedekind domain is a unique factorization domain (UFD) iff it is a PID.


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 m. In the attempt to determine which integers are represented by the quadratic form x2 + my2, it is natural to factor the quadratic form into (x+\sqrt{-m}y)(x-\sqrt{-m}y), the factorization taking place in the ring of integers of the quadratic field \mathbb{Q}(\sqrt{-m}). Similarly, for a positive integer n the left hand side of the Fermat equation xn + yn = zn can be factored over the ring \mathbb{Z}[\zeta_n], where ζn is a primitive n root of unity.

For a few small values of m and n these rings of algebraic integers are PIDs, and this can be seen as an explanation of the classical successes of Fermat (m = 1,n = 4) and Euler (m = 2,3,n = 3). By this time a procedure for determining whether the ring of all algebraic integers of a given quadratic field \mathbb{Q}(\sqrt{D}) 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 D < 0 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 n > 2 -- 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 \mathbb{Z}[\zeta_n] is a UFD. Ernst Kummer had shown three years before that this was not the case already for n = 23 (the full, finite list of values for which \mathbb{Z}[\zeta_n] 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 n using what we now recognize as the fact that the ring \mathbb{Z}[\zeta_n] 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.

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