Editing Dedekind domain (section)

== Some examples of Dedekind domains ==

All principal ideal domains and therefore all discrete valuation rings are Dedekind domains.

The ring <math>R = \mathcal{O}_K</math> of [[algebraic integers]] in a number field ''K'' is Noetherian, integrally closed, and of dimension one: to see the last property, observe that for any nonzero prime ideal ''I'' of ''R'', ''R''/''I'' is a finite set, and recall that a finite integral domain is a field; so by (DD4) ''R'' is a Dedekind domain.  As above, this includes all the examples considered by Kummer and Dedekind and was the motivating case for the general definition, and these remain among the most studied examples.

The other class of Dedekind rings that is arguably of equal importance comes from geometry: let ''C'' be a nonsingular geometrically integral ''[[Affine variety#Affine varieties|affine]]'' [[algebraic curve]] over a field ''k''.  Then the [[coordinate ring]] ''k''[''C''] of regular functions on ''C'' is a Dedekind domain.  This is largely clear simply from translating geometric terms into algebra:  the coordinate ring of any affine variety is, by definition, a finitely generated ''k''-algebra, hence Noetherian; moreover ''curve'' means ''dimension one'' and ''nonsingular'' implies (and, in dimension one, is equivalent to) ''normal'', which by definition means ''integrally closed''.

Both of these constructions can be viewed as special cases of the following basic result:

'''Theorem''': Let ''R'' be a Dedekind domain with [[field of fractions|fraction field]] ''K''.  Let ''L'' be a finite degree [[field extension]] of ''K'' and denote by ''S'' the [[integral closure]] of ''R'' in ''L''.  Then ''S'' is itself a Dedekind domain.<ref>The theorem follows, for instance, from the [[Krull–Akizuki theorem]].</ref>

Applying this theorem when ''R'' is itself a PID gives us a way of building Dedekind domains out of PIDs.  Taking ''R'' = '''Z''', this construction says precisely that rings of integers of number fields are Dedekind domains.  Taking ''R'' = ''k''[''t''], one obtains the above case of nonsingular affine curves as [[branched covering]]s of the affine line.

[[Oscar Zariski|Zariski]] and [[Pierre Samuel|Samuel]] were sufficiently taken with this construction to ask whether every Dedekind domain arises from it; that is, by starting with a PID and taking the integral closure in a finite degree field extension.<ref>Zariski and Samuel, p. 284</ref>  A surprisingly simple negative answer was given by L. Claborn.<ref>Claborn 1965, Example 1-9</ref>

If the situation is as above but the extension ''L'' of ''K'' is algebraic of infinite degree, then it is still possible for the integral closure ''S'' of ''R'' in ''L'' to be a Dedekind domain, but it is not guaranteed.  For example, take again ''R'' = '''Z''', ''K'' = '''Q''' and now take ''L'' to be the field <math>\overline{\textbf{Q}}</math> of all algebraic numbers.  The integral closure is then the ring <math>\overline{\textbf{Z}}</math> of all algebraic integers.  Since the square root of an algebraic integer is again an algebraic integer, it is not possible to factor any nonzero nonunit algebraic integer into a finite product of irreducible elements, which implies that <math>\overline{\textbf{Z}}</math> is not even Noetherian.  In general, the integral closure of a Dedekind domain in an infinite algebraic extension is a [[Prüfer domain]]; it turns out that the ring of algebraic integers is slightly more special than this: it is a [[Bézout domain]].