---
tag: WNQDKQYZ2R
kind: concept
name: Galois theory
msc: [12F10]
sources:
  - url: https://en.wikipedia.org/wiki/Galois_theory
same_as:
  wikidata: Q92552
  wikipedia: Galois theory
---

## Summary

Galois theory studies a field extension through its group of automorphisms.
Its central result, the [[fundamental theorem of Galois theory|0T1Q8D0FW8]], matches the intermediate fields of a finite [[Galois extension|0DS2F7GK05]] with the subgroups of its [[Galois group|0XVR72M51M]].
It explains which polynomial equations are [[solvable by radicals|J7YS0DA3HQ]] and which constructions with straightedge and compass are possible.

## Explanations

### L1 human

The roots of a polynomial can often be permuted without breaking any algebraic relation among them.
These permutations form a group, and Galois theory reads properties of the equation off this group.
For $x^2 - 2$, the two roots $\sqrt{2}$ and $-\sqrt{2}$ can be exchanged, and the group has two elements.
Over a field of characteristic $0$, a polynomial can be solved with roots and the four basic operations exactly when its group is [[solvable|0P3KZK0FTV]].
This is why there is a formula for degree $4$ and none for the general equation of degree $5$.