Inverse Galois problem

The inverse Galois problem asks which finite groups occur as Galois groups of field extensions, especially whether every finite group occurs over the rational numbers.

Galois group: The group of automorphisms of a field extension that fix its base field. The problem asks which finite groups can arise as these symmetry groups.
Hilbert's irreducibility theorem: A theorem ensuring that many specializations of an irreducible polynomial over a rational function field remain irreducible. Specializing function-field extensions can produce Galois extensions over the rational numbers.
Symmetric group: The group of all permutations of a finite set. Symmetric groups arise as Galois groups of generic polynomials and many specific polynomials.
Rational numbers: The field of ratios of integers. The best-known version asks whether every finite group occurs over this field.
Finite group: A group with finitely many elements. Finite groups are the objects whose realization as Galois groups is in question.
Regular Galois extension: A Galois extension of a rational function field whose constants are exactly the base field’s constants. Regular extensions often yield number-field realizations through specialization.
Alternating group: The group of even permutations of a finite set. Alternating groups occur as Galois groups and form a major family of simple examples.
Abelian extension: A Galois extension whose Galois group is abelian. Class field theory gives a comprehensive account of abelian realizations over number fields.
Galois extension: A field extension that is normal and separable. Its automorphism group provides the Galois group whose possible values the problem studies.
Hilbert's theorem 90: A theorem describing elements of norm one in cyclic Galois extensions as quotients of conjugates. Its cohomological form helps classify extensions and solve embedding problems.

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