Regular Galois Extension

A regular Galois extension over a field is a finite Galois extension in which every algebraic element of over already belongs to . A regular -extension is a regular Galois extension whose Galois group is isomorphic to a specified finite group .

If a nontrivial finite group has a regular -extension over , Hilbert's irreducibility theorem implies that specializing to suitable values in gives infinitely many -extensions over . Huang et al. (2026) constructed such an extension for the Mathieu group M₂₃, so every one of the 26 sporadic groups has a regular Galois extension over .