Dedekind domain

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 that are sometimes taken as the definition: see below. 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 that are sometimes taken as the definition: see below. 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) if and only if it is a PID. In the 19th century it became a common technique to gain insight into integral solutions of polynomial equations using rings of algebraic numbers of higher degree. For instance, fix a positive integer m {displaystyle m} . In the attempt to determine which integers are represented by the quadratic form x 2 + m y 2 {displaystyle x^{2}+my^{2}} , it is natural to factor the quadratic form into ( x + − m y ) ( x − − m y ) {displaystyle (x+{sqrt {-m}}y)(x-{sqrt {-m}}y)} , the factorization taking place in the ring of integers of the quadratic field Q ( − m ) {displaystyle mathbb {Q} ({sqrt {-m}})} . Similarly, for a positive integer n {displaystyle n} the polynomial z n − y n {displaystyle z^{n}-y^{n}} (which is relevant for solving the Fermat equation x n + y n = z n {displaystyle x^{n}+y^{n}=z^{n}} ) can be factored over the ring Z [ ζ n ] {displaystyle mathbb {Z} } , where ζ n {displaystyle zeta _{n}} is a primitive n {displaystyle n} root of unity. For a few small values of m {displaystyle m} and n {displaystyle 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 {displaystyle m=1,n=4} ) and Euler ( m = 2 , 3 , n = 3 {displaystyle m=2,3,n=3} ). By this time a procedure for determining whether the ring of all algebraic integers of a given quadratic field Q ( D ) {displaystyle 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 {displaystyle 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 Kurt Heegner, Alan Baker and Harold 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 {displaystyle 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 Z [ ζ n ] {displaystyle mathbb {Z} } is a UFD. Ernst Kummer had shown three years before that this was not the case already for n = 23 {displaystyle n=23} (the full, finite list of values for which Z [ ζ n ] {displaystyle mathbb {Z} } 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 {displaystyle n} using what we now recognize as the fact that the ring Z [ ζ n ] {displaystyle mathbb {Z} } 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 O K {displaystyle {mathcal {O}}_{K}} of algebraic integers in a number field K {displaystyle K} need not be a PID. In fact, although Gauss also conjectured that there are infinitely many primes p {displaystyle p} such that the ring of integers of Q ( p ) {displaystyle mathbb {Q} ({sqrt {p}})} is a PID, as of 2016 we do not even know whether there are infinitely many number fields K {displaystyle K} (of arbitrary degree) such that O K {displaystyle {mathcal {O}}_{K}} 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 R {displaystyle R} is Dedekind iff for every maximal ideal M {displaystyle M} of R {displaystyle R} the localization R M {displaystyle R_{M}} 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. For an integral domain R {displaystyle R} that is not a field, all of the following conditions are equivalent: