# Ideal class group isomorphic to $\mathbb{Z}$

Hi everybody,

I wonder if someone could provide me with a simple example of a Dedekind ring whose ideal class group is isomorphic to $\mathbb{Z}$.

The point is I would like the example simple enough to be understood by a 4^th year student, with basic knowledge in algebra.

I know that every abelian group is the ideal class group of some Dedekind ring. It is proved in a paper of Claborn. I've read through the paper, and followed the proof, but it seems very intricate and needs lots of prerequisites (it needs stuff on Krull domains, valuation rings, plus other intermediate results)

In fact, any example of a Dedekind ring with an infinite ideal class group would do. I've seen in an other topic that the integral closure of $\mathbb{C}[X]$ in $\mathbb{C}(X)(\sqrt{X^3-X})$ would do, but the proof needs some non trivial results on elliptic curves.

I really would like the example and the computation of the ideal class group as self-contained as it is possible. Maybe I'm asking too much, I don't know.

Thanks in advance !

Greg

• Claborn's proof can be found here: ttp://www.math.uga.edu/~pete/claborn66.pdf Have you tried to work it out in the special case $\mathbb{Z}$? – Martin Brandenburg May 1 '12 at 9:46
• The example you mention can not have class group $\mathbb{Z}$. In fact, for any finitely generated algebra over $\mathbb{C}$, which is a Dedekind domain, the class group is divisible and hence can not be the integers. If you want to work with affine algebras over algebraically closed fields, you have to get out of Dedekind rings. Why not the easy universal one $\mathbb{C}[x,y,z,w]/(xy-zw)$? – Mohan May 1 '12 at 10:08
• @GreginGre: Yes, but perhaps you can write down the ring explicitly (sure you can) in this special case and then start your own more elementary proof on it (if possible). – Martin Brandenburg May 1 '12 at 10:29
• Why don't you use an ell. curve over Q (not over C) whose group of rational points is isom. to Z? There are plenty of examples of that. – KConrad May 1 '12 at 13:00
• You can give an example of infinite class group without explicitly using algebraic geometry, if the students have seen elliptic functions (a.k.a. doubly periodic functions), as treated in Ahlfors's "Complex Analysis". – Tom Goodwillie May 1 '12 at 16:53