  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.

Number Theory;Rings and Algebras