= Frobenius automorphism
{c}
{title2=$\operatorname{Frob}$}
= Frobenius
{c}
{synonym}
= Arithmetic Frobenius
{synonym}
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 $\operatorname{Gal}(L/K)$. 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 $\mathbb F_p$.
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