When is a field extension Galois?

A finite extension is Galois when it is both normal (it contains all conjugates of each of its elements) and separable (minimal polynomials have distinct roots). Equivalently, the automorphism group fixing the base has order equal to the degree.

Why view the Galois group as permuting roots?

An automorphism fixing the base field must send roots of a polynomial to other roots, so the group acts on the finite set of roots. This realizes the Galois group inside a symmetric group, making it computable and connecting it to permutation-group theory.

Galois Group

The Galois group of a field extension is the group of field automorphisms fixing the base field, encoding the symmetries of the roots of a polynomial and indexing the intermediate fields.

Definition

For a field extension, the Galois group is the group of automorphisms of the larger field that fix every element of the base field; the extension is called Galois when this group is as large as the degree, which happens exactly for finite normal and separable extensions.

Scope

This topic covers automorphisms of field extensions, the definition of the Galois group, normal and separable extensions, the fundamental theorem of Galois theory, and the computation of Galois groups of polynomials and their interpretation as permutation groups of roots.

Core questions

What symmetries does a field extension possess?
When is an extension Galois, and how large is its automorphism group?
How does the Galois group correspond to intermediate fields?
How is the Galois group of a polynomial realized as a permutation group of its roots?

Key theories

Fundamental theorem of Galois theory: For a finite Galois extension there is an inclusion-reversing bijection between intermediate fields and subgroups of the Galois group, under which the degree of a subextension equals the index of the corresponding subgroup.
Galois group as permutations of roots: The Galois group of a separable polynomial acts faithfully on its roots, embedding it as a subgroup of the symmetric group on those roots, which constrains and helps compute the group.
Artin's theorem on fixed fields: If a finite group of automorphisms acts on a field, the whole field is a Galois extension of the fixed subfield with that group as its Galois group, giving a converse to the construction of Galois groups.

Clinical relevance

The Galois group converts questions about field extensions and polynomial equations into group theory; its solvability decides solvability by radicals, and the inverse Galois problem and Galois representations make it central to modern number theory and arithmetic geometry.

History

Galois associated to each equation a group of permutations of its roots in the 1830s, the original Galois group. Dedekind and Artin recast this in terms of automorphisms of fields, and Artin's formulation in terms of fixed fields gave the theory its modern, conceptual shape.

Key figures

Évariste Galois
Emil Artin
Richard Dedekind
Leopold Kronecker

Seminal works

dummit2004
lang2002
artin2011

Frequently asked questions

When is a field extension Galois?: A finite extension is Galois when it is both normal (it contains all conjugates of each of its elements) and separable (minimal polynomials have distinct roots). Equivalently, the automorphism group fixing the base has order equal to the degree.
Why view the Galois group as permuting roots?: An automorphism fixing the base field must send roots of a polynomial to other roots, so the group acts on the finite set of roots. This realizes the Galois group inside a symmetric group, making it computable and connecting it to permutation-group theory.