Galois group

The group of field automorphisms of an extension that fix every element of its base field. For a finite Galois extension, its size equals the extension’s degree.

Field automorphism: A bijective map from a field to itself that preserves addition, multiplication, and the identity element. Each element of a Galois group is an automorphism constrained to fix the base field.
Fundamental theorem of Galois theory: The theorem matching intermediate fields of a finite Galois extension with subgroups of its Galois group. It converts group structure into a complete description of intermediate fields.
Field extension: An inclusion of fields, viewed as a larger field containing a smaller base field. A Galois group is defined relative to this inclusion.
Field embedding: An injective homomorphism from one field into another that preserves field operations and identity. An embedding need not map an extension onto itself, unlike a group automorphism.
Group action: A rule assigning each group element a transformation of a set, compatibly with group multiplication. Galois group elements act on extension elements and, especially, polynomial roots.
Solvability by radicals: The possibility of expressing polynomial roots using field operations and extraction of roots. A polynomial over a characteristic-zero field is solvable by radicals precisely when its Galois group is solvable.
Fixed field: The subfield consisting of elements left unchanged by every automorphism in a specified group. The base field is the fixed field of the full Galois group in a finite Galois extension.
Automorphism group: The group of all automorphisms of a mathematical structure under composition. A Galois group is the subgroup of field automorphisms that fixes a chosen base field.
Splitting field: The smallest field extension of a given field in which a polynomial splits into linear factors. Its automorphisms permute all roots while preserving their algebraic relations.
Abelian extension: A Galois extension whose Galois group is an abelian group. Commutative symmetry leads to especially tractable field extensions and reciprocity laws.

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