---
tag: YJS6X8JN7D
kind: problem
name: Inverse Galois problem
msc: [12F12, 11R32]
uses: [0DS2F7GK05, 0XVR72M51M]
about: [0XVR72M51M]
sources:
  - url: https://en.wikipedia.org/wiki/Inverse_Galois_problem
same_as:
  wikidata: Q2358071
  wikipedia: Inverse Galois problem
history:
  posed_by: "David Hilbert"
---

## Problem

Is every finite group $G$ isomorphic to the [[Galois group|0XVR72M51M]] $\Gal(L/\Q)$ of some [[Galois extension|0DS2F7GK05]] $L$ of the rational numbers $\Q$?

## Remarks

The problem is open.
Every finite abelian group is a Galois group over $\Q$, realised inside a [[cyclotomic extension|VD6VC3ZMMY]] $\Q(\zeta_n)$.
The symmetric groups $S_n$ are Galois groups over $\Q$; one route uses [[Hilbert's irreducibility theorem|N1V3YEYHNP]].
Every finite solvable group is a Galois group over $\Q$, by [[Shafarevich's theorem|SDV692Z915]].
The same question can be asked over other fields; over $\C(t)$ every finite group is a Galois group.
Over a finite field $\F_q$ only cyclic groups occur, since every finite extension of $\F_q$ is cyclic.

## Explanations

### L1 human

Galois theory attaches a finite group to each polynomial with rational coefficients.
The inverse problem asks whether every finite group appears in this way.
Many families of groups are known to appear, but no proof covers all finite groups.