---
tag: 0VP4TBVJ19
kind: definition
name: Frobenius automorphism of a finite field
msc: ["12E20"]
defines: 0CGCPX8Z70
uses: [0GHDDVAH8P, 08RTDHX1SK, 0B6QC0C0G1]
sources:
  - url: https://en.wikipedia.org/wiki/Finite_field
    locator: Frobenius automorphism and Galois theory
  - url: https://en.wikipedia.org/wiki/Frobenius_endomorphism
    locator: lead section
---

## Definition

Let $F$ be a [[finite field|08RTDHX1SK]] of characteristic $p$.
The Frobenius automorphism of $F$ is the map $\varphi \colon F \to F$, $\varphi(x) = x^p$.

## Remarks

The map $\varphi$ is the [[Frobenius endomorphism|0GHDDVAH8P]] of $F$. It is injective, as is every field homomorphism, so it is bijective, because $F$ is finite.
The map $\varphi$ fixes every element of the prime subfield $\mathbb{F}_p$, by Fermat's little theorem.
If $|F| = p^n$, then $\varphi^n$ is the identity, and the automorphisms of $F$ are exactly $\mathrm{id}, \varphi, \dots, \varphi^{n-1}$.
So $\mathrm{Gal}(\mathbb{F}_{p^n}/\mathbb{F}_p)$ is cyclic of order $n$, generated by $\varphi$.

## Explanations

### L1 human

Raising to the power $p$ respects addition in characteristic $p$, by the freshman's dream.
In a finite field, this map permutes the elements, and every symmetry of the field is a power of it.