On the Infinitude of Prime Ideals in Dedekind Domains

Let $R$ be an infinite Dedekind domain with at most finitely many units, and let $K$ denote its field of fractions. We prove the following statement. If $L/K$ is a finite Galois extension of fields and $mathcal{O}$ is the integral closure of $R$ in $L$, then $mathcal{O}$ contains infinitely many prime ideals. In particular, if $mathcal{O}$ is further a unique factorization domain, then $mathcal{O}$ contains infinitely many non-associate prime elements.