How to construct rings with a given class number?

Question

Hi I was learning about class number and I was wondering if it is known how to construct rings for any specific class number.

Answer 1

Yes, for every finite abelian group one can construct a Dedekind domain with class group isomorphic to that group. This is a result from the 1960s due to Claborn.

For details see for example:

Pete L. Clark: Elliptic Dedekind domains revisited. Enseignement Math. 55 (2009), 213-225.

Answer 2

If you allow me to interpret your question for more general rings than Dedekind rings, then there are also the following examples. Let $C\subseteq \mathbf P^2$ be a nonsingular curve of degree $h$, Then the complement $\mathbf P^2\setminus C$ is an affine algebraic variety whose coordinate ring has class group $\mathbf Z/h$.