---
tag: 0JZY3R908H
kind: definition
name: Galois group of a polynomial
msc: [12F10]
defines: 0Q5NA8Y362
uses: [0XVR72M51M, 02NZZ1J0FH]
sources:
  - url: https://en.wikipedia.org/wiki/Galois_group
    locator: Galois group of a polynomial
---

## Definition

Let $K$ be a [[field|0QA8CHH8MG]] and let $f \in K[x]$ be nonzero.
Let $L$ be a [[splitting field|02NZZ1J0FH]] of $f$ over $K$.
The Galois group of $f$ over $K$ is the group $\operatorname{Aut}(L/K)$ of the [[automorphisms of $L/K$|0XVR72M51M]].

## Remarks

By the [[uniqueness of splitting fields|069A3BC3EP]], the group does not depend on the choice of $L$, up to isomorphism.
Each automorphism permutes the distinct roots $r_1, \dots, r_m$ of $f$ in $L$, and it is determined by this permutation, because the roots generate $L$. So the group is isomorphic to a subgroup of the symmetric group $S_m$.
The Galois group of $x^2 - 2$ over $\mathbb{Q}$ has order $2$. The Galois group of $x^3 - 2$ over $\mathbb{Q}$ is isomorphic to $S_3$.
If $f$ is separable, then $L/K$ is a [[Galois extension|0DS2F7GK05]].

## Explanations

### L1 human

The Galois group of a polynomial lists the ways to permute its roots that respect all the algebraic relations between them.
It measures how complicated the polynomial is.
For example, a polynomial over $\mathbb{Q}$ can be solved by radicals exactly when its Galois group is solvable.