Editing Galois theory (section)

{{short description|Mathematical connection between field theory and group theory}}
[[File:Lattice diagram of Q adjoin the positive square roots of 2 and 3, its subfields, and Galois groups.svg|alt=Lattice of subgroups and subfields showing their corresponding Galois groups.|thumb|400x400px|On the left, the [[Lattice (order theory)|lattice]] diagram of the field obtained from {{math|'''Q'''}} by adjoining the positive square roots of 2 and 3, together with its subfields; on the right, the corresponding lattice diagram of their Galois groups.]]
In [[mathematics]], '''Galois theory''', originally introduced by [[Évariste Galois]], provides a connection between [[field (mathematics)|field theory]] and [[group theory]]. This connection, the [[fundamental theorem of Galois theory]], allows reducing certain problems in field theory to group theory, which makes them simpler and easier to understand.

Galois introduced the subject for studying [[root of a function|root]]s of [[polynomial]]s. This allowed him to characterize the [[polynomial equation]]s that are '''solvable by radicals''' in terms of properties of the [[permutation group]] of their roots—an equation is by definition ''solvable by radicals'' if its roots may be expressed by a formula involving only [[integer]]s, [[nth root|{{mvar|n}}th roots]], and the four basic [[arithmetic operations]]. This widely generalizes the [[Abel–Ruffini theorem]], which asserts that a general polynomial of degree at least five cannot be solved by radicals. 

Galois theory has been used to solve classic problems including showing that two problems of antiquity cannot be solved as they were stated ([[doubling the cube]] and [[trisecting the angle]]), and characterizing the [[regular polygon]]s that are [[Constructible polygon|constructible]] (this characterization was previously given by [[Gauss]] but without the proof that the list of constructible polygons was complete; all known proofs that this characterization is complete require Galois theory).

Galois' work was published by [[Joseph Liouville]] fourteen years after his death. The theory took longer to become popular among mathematicians and to be well understood.

Galois theory has been generalized to [[Galois connection]]s and [[Grothendieck's Galois theory]].