The Inverse Galois Problem is a central question in the field of algebra, particularly in the area of field theory and algebraic geometry. It seeks to determine whether every finite group can be represented as the Galois group of some field extension of the rational numbers \(\mathbb{Q}\) or more generally, of some base field.
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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 . A regular realization over 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 .