A Dedekind Domain with Nontrivial Class Group

Abstract

Analytic properties of function spaces over the real and the complex fields are different in some ways. This reflects in algebraic properties which are different at times and similar in some other respects. For instance, the ring of real-valued continuous functions on a closed interval like $[0,1]$ behaves similarly to the corresponding ring of complex-valued functions; they depend only on the topology of $[0,1]$. The ring $\mathbf{R}[X,Y]/(X^2+Y^2-1)$ of real-valued polynomial functions on the unit circle is not a unique factorization domain - witness the equation $$\cos^2(t) = (1+ \sin(t))(1- \sin(t)).$$ On the other hand, the ring $\mathbf{C}[X,Y]/(X^2+Y^2-1) \cong \mathbf{C}[X+iY, 1/(X+iY)]$ is a principal ideal domain. Again, the rings of convergent power series (over either of these fields) with radius of convergence larger than some number $ρ$ is a Euclidean domain (and hence, a principal ideal domain) - this can be seen by using for a Euclidean "norm" function, the function which counts zeroes (with multiplicity) in the disc $|z| \leq ρ$. In this note, we consider the rings $C_{an}(S^1;\mathbf{R})$ of real-analytic functions on the unit circle $\mathit{S}^1$ which are real-valued and the corresponding ring $C_{an}(S^1; \mathbf{C})$ of analytic functions that are complex-valued. We will see that the latter is a principal ideal domain while the former is a Dedekind domain which is not a principal ideal domain - the class group having order $2$.