If $L|K$ is a Galois extension of algebraic number fields, and $\mathfrak{P}$ a prime ideal which is unramified over $K$ (i.e. $\mathfrak{p} = \mathfrak{P} \cap K$ is unramified in $L$), then there is one and only one automorphism $\phi_{\mathfrak{P}} \in G(L|K)$ such that
$\phi_{\mathfrak{P}}a \simeq a^q \ mod \ \mathfrak{P}$ for all $a \in \mathcal{O}$,
where $q = [\kappa(\mathfrak{P}) : \kappa(\mathfrak{p})]$. It is called the Frobenius automorphism. The decomposition group $G_{\mathfrak{P}}$ is cyclic and $\phi_{\mathfrak{p}}$ is a generator of $G_{\mathfrak{P}}$.
Typo: That should be $q = |\kappa(\mathfrak{p})|.$