---
tag: MP5C9T2EYG
kind: statement
role: theorem
name: Galois group of a finite field extension
msc: ["12E20", 12F10]
uses: [08RTDHX1SK, 0DS2F7GK05, 0XVR72M51M, 0B85Q41RP6, 0VP4TBVJ19, 0QWR8WTZEW, 0QM98DNWVW, 0T1Q8D0FW8, 08YERZDM9X]
about: [0XTY86BZKY, 0ZXZ82FTX2]
sources:
  - url: https://en.wikipedia.org/wiki/Finite_field
    locator: Frobenius automorphism and Galois theory
---

## Statement

Let $q$ be a prime power, let $n \ge 1$, and let $\F_q \subseteq \F_{q^n}$ be [[finite fields|08RTDHX1SK]] of orders $q$ and $q^n$.
Then $\F_{q^n}$ is a [[Galois extension|0DS2F7GK05]] of $\F_q$.
Its [[Galois group|0XVR72M51M]] is [[cyclic|0B85Q41RP6]] of order $n$, generated by the map $\varphi(x) = x^q$.

## Remarks

The map $\varphi$ is a power of the [[Frobenius automorphism|0VP4TBVJ19]], so it is a field automorphism of $\F_{q^n}$.
It fixes $\F_q$, because [[every element of $\F_q$ satisfies $a^q = a$|0QWR8WTZEW]].
If $\varphi^k = \id$ with $k \ge 1$, then every element of $\F_{q^n}$ is a root of $x^{q^k} - x$, so $q^n \le q^k$ and $k \ge n$; and $\varphi^n = \id$.
So $\varphi$ has order $n$, and since [[the Galois group has order at most the degree|0QM98DNWVW]] $n$, the group is $\langle \varphi \rangle$ and the extension is Galois.
For example, $\Gal(\F_4/\F_2) = \set{\id, \varphi}$ with $\varphi(x) = x^2$, and $\varphi$ swaps the two roots $\omega$ and $\omega + 1$ of $x^2 + x + 1$.

## Explanations

### L1 human

A finite field has very few symmetries over a smaller finite field.
All of them are powers of one map, raising to the $q$-th power.
So the symmetry group is as simple as possible: a cycle of length $n$.

### L2 human

With the [[fundamental theorem of Galois theory|0T1Q8D0FW8]], the subgroups of a cyclic group of order $n$ match the fields between $\F_q$ and $\F_{q^n}$.
This gives again that these fields are the $\F_{q^m}$ with $m$ dividing $n$, as in [[subfields of a finite field|08YERZDM9X]].
The absolute Galois group of $\F_q$ is the profinite completion $\widehat{\Z}$ of $\Z$, topologically generated by $\varphi$.
The Frobenius element of a prime in algebraic number theory comes from this theorem applied to residue fields.