Frobenius order is residue degree

Given: The data and hypotheses of the statement.

Arithmetic frobenius coset: For finite Galois L/K and nonzero $P\mid p$, the arithmetic Frobenius coset is the unique element of $D(P\mathrm{/}p)\mathrm{/}I(P\mathrm{/}p)$ corresponding under the residue isomorphism to $x\mapsto x^{Np}$ on $\kappa (P)$, where $Np=\mathrm{\mid }\kappa (p)\mathrm{\mid }$. It is defined also when P is ramified. Its inverse is called geometric Frobenius. The quotient element is distinguished; a representative in D need not be unique.

Unramified frobenius element exists uniquely: For finite Galois L/K and a nonzero prime $P\mid p$ with $e(P\mathrm{/}p)=1$, there is a unique ${\mathrm{Frob}}_P\in D(P\mathrm{/}p)$ satisfying ${\mathrm{Frob}}_P(a)\equiv a^{Np}(\mathrm{m}\mathrm{o}\mathrm{d}P)(a\in {\mathcal{O}}_L)\mathrm{.}$ It is the arithmetic Frobenius element, the unique lift of the arithmetic Frobenius coset.

A finite extension of a finite field of order $q$ is Galois with cyclic Galois group generated by $x\mapsto x^q$: Let ${\mathbb{F}}_q$ be a finite field of order $q$ and let $E$ be a finite field having ${\mathbb{F}}_q$ as a subfield, with $[E:{\mathbb{F}}_q]=n$ (def-extension-degree-and-finite-extension). Then $E\mathrm{/}{\mathbb{F}}_q$ is a finite Galois extension (def-finite-galois-extension-and-galois-group) and $\mathrm{Gal}(E\mathrm{/}{\mathbb{F}}_q)=⟨{\sigma }_q⟩$ is cyclic of order $n$, generated by the relative Frobenius ${\sigma }_q\text{\hspace{0.17em}⁣}:x\mapsto x^q$ (def-relative-frobenius-of-a-finite-field-extension).