The Frobenius Automorphism for a Finite Field

# The Frobenius Automorphism for a Finite Field

 Definition: Let $F$ be a finite field with $\mathrm{char}(F) = p$. The Frobenius Automorphism on $F$ is the automorphism $\phi : F \to F$ defined for all $x \in F$ by $\phi(x) = x^p$.
 Theorem 1: Let $K$ be a finite field with $\mathrm{char}(F) = p$ and with $|K| = p^r$ and let $F$ be an extension field of $F$ such that $[F:K] = m$. Then: a) $\mathrm{Gal} (F/K)$ is isomorphic to the cyclic group of order $m$. b) $\mathrm{Gal} (F/K)$ is generated by $\phi^r$ where $\phi : F \to F$ is the Frobenius automorphism.
• Proof of a): Since $K$ is a finite field and $[F:K] = m$, $F$ is a finite extension of $K$ and:
(1)
\begin{align} \quad |F| = (p^r)^m = p^{rm} \end{align}
• Therefore $F$ is the splitting field of the polynomial:
(2)
\begin{align} \quad f(x) = x^{p^{rm}} - x \end{align}
• Over the prime subfield of $F$ which is $K$. But $f(x)$ has no repeated roots and so:
(3)
\begin{align} \quad |\mathrm{Gal}(F/K)| = [F:K] = m \quad \blacksquare \end{align}