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Concept
Dedekind domain
Summary
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 integer solutions of polynomial 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 n-th 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 were no further values.
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