# Inverse Galois problem

- kind: problem
- status: open
- year: 1850

**Statement.** Is every finite group the Galois group of some extension of the rational numbers?

Known for solvable groups (Shafarevich) and for all but one of the sporadic groups; the Mathieu group M23 is the famous holdout. Most methods use rigidity, building extensions from the geometry of covering spaces.

Posed 19th century; systematic work from Hilbert and Emmy Noether.

## What would settle it

Lean statement in formal-conjectures

## Sources

- [Wikipedia: Inverse Galois problem](https://en.wikipedia.org/wiki/Inverse_Galois_problem)

[Algebra and groups](/wiki/Space:algebra.md) · [Mathematics](/wiki/Continent:mathematics.md)

## Relations to this page

- related to [quintic-insolvability](/wiki/Problem:quintic-insolvability.md) — The original curation links this problem to Inverse Galois problem; the argument for the link has not been written yet.

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