 Frobenius automorphism

Let $K$ be a non-Archimedean local field with residue field $k_K$ of cardinality $q$. Arithmetic Frobenius automorphism acts on the algebraic closure of the residue field by $x\mapsto x^q$. On $k_K$ itself this is the identity; on a degree-$f$ residue extension it has order $f$.

For the degree-$f$ unramified extension $L/K$, reduction identifies its Galois group with the residue-field Galois group, so there is a unique lifted automorphism with this residue action. It generates $\mathrm{Gal}(L/K)$. Geometric Frobenius is its inverse. For example, in an unramified quadratic extension of ${\mathbb{Q}}_p$, arithmetic Frobenius lifts $x\mapsto x^p$ on ${\mathbb{F}}_{p^2}$, rather than acting nontrivially on ${\mathbb{F}}_p$.