Thus a Dedekind domain is a domain that satisfies any one, and hence all five, of ( DD1 ) through ( DD5 ).
22.
*PM : ideal decomposition in Dedekind domain, id = 7219-- WP guess : ideal decomposition in Dedekind domain-- Status:
23.
*PM : ideal decomposition in Dedekind domain, id = 7219-- WP guess : ideal decomposition in Dedekind domain-- Status:
24.
However there is a unique factorization for ideals, which is expressed by the fact that every ring of algebraic integers is a Dedekind domain.
25.
Now we can appreciate ( DD3 ) : in a Dedekind domain and only in a Dedekind domain !-- is every fractional ideal invertible.
26.
The name " divisor " goes back to the work of Weber, who showed the relevance of Dedekind domains to the study of algebraic curves.
27.
Taking " R " = " Z " this construction tells us precisely that rings of integers of number fields are Dedekind domains.
28.
Remarkably, the additional structure in torsionfree finitely generated modules over an arbitrary Dedekind domain is precisely controlled by the class group, as we now explain.
29.
For a Dedekind domain, this is the case : indeed, K 1 is generated by the images of GL 1 and SL 2 in GL.
30.
The integral closure of a Dedekind domain in a finite extension of the field of fractions is a Dedekind domain; in particular, a noetherian ring.
