##  [Galois Theory](/galois-theory-0) 

 Definition

The study of the relationship between field extensions and groups of field automorphisms that permute algebraic roots, giving a correspondence between intermediate fields and subgroups of an automorphism group, and linking solvability questions to group-theoretic properties.

 

 

 

 

 

 





## Principle

Principle

Normality and separability of an extension produce a group of automorphisms whose subgroup lattice reflects the lattice of intermediate fields; the Galois correspondence associates fixed fields to subgroups and subgroups to fixed fields, organizing extension structure by symmetry.

 

 

 

 

 





## Demonstration

Demonstration

Classical demonstration: for a finite separable normal extension E/F (a Galois extension), there is a bijection between intermediate fields K (F ⊂ K ⊂ E) and subgroups H ≤ Gal(E/F), given by K ↦ Gal(E/K) and H ↦ E^H (the fixed field). Concrete scenario: the splitting field of x^3−2 over Q has Galois group S_3 and intermediate fields corresponding to subgroups, which explains solvability by radicals or its failure.

 

 

 

 

## Misapplication

Misapplication

Assuming every algebraic extension is Galois and so has a clean subgroup–field correspondence; ignoring inseparability in positive characteristic; or treating the Galois group merely as permutations of listed roots without considering field-structural automorphisms and topology in the infinite case.

 

 

 

 

 





## Consequence

Consequence

When applicable, Galois theory reduces field-theoretic problems to group theory: classification of extensions, determination of solvability by radicals, explicit construction of resolvents, and use of cohomological invariants for obstruction problems. It also motivates profinite topology in infinite Galois theory.

 

 

 

 

## Reversal

Reversal

The reversal emphasizes reconstructing field data from group- or cohomological data: given a profinite group with extra structure one tries to realize it as a Galois group of some field; inversion highlights limitations and rigidity problems (inverse Galois problem).

 

 

 

 

 





## Boundary

Boundary

Primarily concerns separable normal (Galois) extensions; finite Galois theory is simplest, while infinite extensions require profinite groups and Krull topology. Excludes non-separable extensions without further structure, and the classical correspondence can fail or require modification in positive characteristic or for non-normal extensions.

 

 

 

 

 





## Semantic Tension

Semantic Tension

Tension between viewing Galois groups as permutation groups on explicit root sets versus as abstract automorphism groups with profinite topology. There is also a nearby meaning in differential Galois theory or geometric/étale analogues where the objects and symmetries differ though the organizing idea persists.

 

 

 

 

 





## Synthesis

Synthesis

Galois theory ties field extensions to symmetry groups: when normality and separability hold, intermediate fields correspond to subgroups of automorphisms, allowing algebraic problems to be translated into group-theoretic and cohomological terms and guiding both constructive solutions and obstructions.