# How many prime ideals are fixed by a given permutation?

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.

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Yes. Let $G$ be the subgroup of the Galois group generated by $\sigma$, and let $K$ be the fixed field of $G$. Then $L/K$ is cyclic, and by Chebotarev there are infinitely many prime ideals in $K$ that are inert in $L$.