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inverse Galois problem

Known for:

• \(S_n\) and \(A_n\), due to Hilbert
• \(\operatorname{PGL}_n\)?
• Any finite abelian group, by the Kronecker-Weber theorem.
• Any solvable group, due to Shaferevich
• The monster group.

Techniques: by Hilbert irreducibility, 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 \(p\in {\mathbf{C}}(t)\), and try to impose enough conditions to ensure \(p\) can be defined over \({\mathbf{Q}}(t)\).