Suppose $L$ is a finite Galois field extension of the rational number field $\mathbb{Q}$, and $B$ is the integral closure of $\mathbb{Z}$ in $L$. Let $\sigma$ be an element of the Galois group Gal($L/\mathbb{Q}$). My question is
Are there infinitely many prime ideals of $B$ which are fixed by $\sigma$, i.e., is the set $\{\mathfrak{p}\in \operatorname{Spec} B\mid \sigma(\mathfrak{p})=\mathfrak{p}\}$ nonempty and infinite?
Thanks.