A Dedekind Domain with Nontrivial Class Group

TL;DR

This paper investigates the algebraic properties of rings of real-analytic and complex-analytic functions on the unit circle, revealing that the complex-valued ring is a PID while the real-valued ring is a Dedekind domain with a nontrivial class group.

Contribution

It demonstrates that the ring of real-analytic functions on the circle is a Dedekind domain with a class group of order 2, contrasting with the complex case which is a PID.

Findings

The complex-analytic function ring is a principal ideal domain.

The real-analytic function ring is a Dedekind domain with a nontrivial class group.

The class group of the real-analytic function ring has order 2.

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 $\rho$ 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 $|z| \leq \rho$. 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 $2$.