Galois theory

Galois theory relates symmetries of polynomial roots, expressed as field automorphisms, to the structure of field extensions. It explains when polynomial equations can be solved using radicals.

Galois group: The group of automorphisms of a field extension that fix its base field. Its structure records the symmetries among roots that Galois theory studies.
Solvability by radicals: The possibility of expressing polynomial roots using field operations and extraction of roots. Galois theory characterizes this possibility through the structure of a polynomial's group.
Field (mathematics): A set with addition, subtraction, multiplication, and division by nonzero elements satisfying familiar arithmetic laws. Galois theory compares fields arranged inside algebraic extensions.
Radical extension: A field extension built through successive adjunctions of roots of elements from smaller fields. A radical extension describes the permitted construction; Galois theory tests whether roots admit one.
Fundamental theorem of Galois theory: The theorem that relates subgroups of a finite Galois group to intermediate fields of its extension. It states the central subgroup–field correspondence behind the theory.
Abel–Ruffini theorem: The theorem that no general formula using radicals solves every polynomial equation of degree five or higher. Its explanation follows from the nonsolvability of the generic quintic's Galois group.
Field extension: An inclusion of fields, viewed as a larger field containing a smaller base field. The theory studies how automorphisms of extensions encode their intermediate fields.
Symmetric group: The group of all permutations of a finite set. Generic polynomials often have full symmetric groups, which obstruct radical solutions in high degree.
Splitting field: The smallest field extension of a base field containing all roots of a given polynomial. Automorphisms of this field capture the polynomial's root symmetries.
Cyclotomic field: A field obtained by adjoining a root of unity to a base field, usually the rational numbers. Cyclotomic extensions give explicit, structured examples of Galois groups and radical solutions.

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