Math 18.786.

· Dedekind domains

I got two questions on Dedekind domains. I only give short answers, which are elaborated with references in a nice Wikipedia article:

http://en.wikipedia.org/wiki/Dedekind_domain

- If A is a PID (which thus also a Dedekind domain) and K=Frac(A) then for any finite extension L of K, the integral closure B of A in L is also a Dedekind domain. (I proved this in class when L/K is separable, but it’s true without this condition.) Can any Dedekind domains be constructed from a PID in this way?

Apparently prestigious mathematicians like Zariski and Samuel asked this question. The answer turns out to be no.

- Can Dedekind domains have infinite groups as ideal class groups?

The answer is yes. In fact it is known that an arbitrary abelian group can be realized as the ideal class group of some Dedekind domain. (So the ring of integers in a number field is special to have the finiteness property, and this fact belongs to number theory rather than an abstract commutative algebra. However it is not the only class of Dedekind domains with finite ideal class groups as every PID has trivial ideal class group. E.g. fields k or k(x).)