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date: 2022-03-23 16:17
modification date: Wednesday 23rd March 2022 16:17:31
title: inverse Galois problem
aliases: [inverse Galois problem]

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# inverse Galois problem

Known for:

- $S_n$ and $A_n$, due to Hilbert
- $\PGL_n$?
- Any finite abelian group, by the [Kronecker-Weber theorem](Unsorted/Kronecker-Weber%20theorem.md).
- Any solvable group, due to Shaferevich
- The [monster group](monster%20group).

Techniques: by [Hilbert irreducibility](Hilbert%20irreducibility), if $G$ occurs as a Galois group over $K(t)$, then it also occurs over $K$.

# Rigidity

Use the fact that every group is the Galois group of some polynomial $p\in \CC(t)$, and try to impose enough conditions to ensure $p$ can be defined over $\QQ(t)$.

![](attachments/Pasted%20image%2020220323163425.png)
![](attachments/Pasted%20image%2020220323163442.png)