Fundamental theorem of Galois theory

For a finite Galois extension, the theorem gives an inclusion-reversing bijection between intermediate fields and subgroups of its Galois group, with fixed fields and automorphism groups paired.

Fixed field: The subfield of a field extension consisting of elements left unchanged by a specified group of automorphisms. Each subgroup corresponds to the field of elements fixed by all its automorphisms.
Field extension: An inclusion of fields in which one field is viewed as containing another. The theorem applies to extensions, with a finite Galois extension as its central setting.
Galois correspondence for finite fields: The explicit correspondence between subfields of a finite field and subgroups of its cyclic Galois group. Finite fields provide a concrete setting where subgroup and subfield lattices can be computed directly.
Infinite Galois theory: The study of possibly infinite Galois extensions using topological groups and closed subgroups. Its correspondence requires a topology and closed subgroups, unlike the finite theorem’s ordinary subgroup lattice.
Galois group: The group of field automorphisms of an extension that fix its base field. Subgroups of this group form one side of the theorem’s correspondence.
Field automorphism: A bijective map from a field to itself that preserves addition, multiplication, and the identity. Automorphisms fixing the base field make up the Galois group.
Cyclotomic field: A field obtained by adjoining a root of unity to the rational numbers. Its Galois group and intermediate fields can be read through the theorem’s subgroup correspondence.
Primitive element theorem: The theorem that every finite separable extension of a field is generated by one element. It describes how an extension is generated, rather than pairing all its intermediate fields with subgroups.
Intermediate field: A field lying between a base field and a larger field extension. These fields form the other side of the correspondence.
Separable extension: An algebraic field extension in which every minimal polynomial has distinct roots. Separability is one of the two defining conditions for a finite Galois extension.

Linked from 7 pages