Inverse Galois problem
= Inverse Galois problem
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The inverse Galois problem asks which finite groups occur as <Galois groups> over a given <field>, with the classical problem taking the base <field> to be $\mathbb Q$. A regular realization over $K(T)$ and <Galois-preserving Hilbert specialization> solve an instance over a <Hilbertian field>. Realization over an algebraically closed constant <field>'s <rational function field> alone does not supply descent to $\mathbb Q$.