A bijective, order-reversing correspondence between intermediate fields of a finite Galois extension and subgroups of its Galois group, realized by mapping a subgroup to its fixed field and an intermediate field to the subgroup of automorphisms that fix it.

Principle

The organizing rule is fixed-field ↔ subgroup: subextensions correspond to subgroups via taking fixed points, and normal subextensions correspond to normal subgroups, giving an anti-isomorphism of lattices in the Galois (finite, separable, normal) case.

Demonstration

Example: For a finite Galois extension K/F, if H ≤ Gal(K/F) then the fixed field K^H is an intermediate extension; conversely each intermediate field E satisfies Gal(K/E) as the corresponding subgroup. Concretely, for quadratic extensions one gets a one-to-one match between the unique intermediate fields and subgroups of order two.

Misapplication

Applying the full bijective correspondence to non-Galois extensions or overlooking separability/normality hypotheses; for instance attempting to match subgroups to intermediate fields for an arbitrary algebraic extension without considering the closure operations leads to incorrect conclusions.

Consequence

Correct application converts field-theoretic problems into group-theoretic ones and vice versa: computation of subextensions, normality checks, and solvability questions reduce to subgroup analysis of the Galois group.

Reversal

In the reversed situation (non-Galois extension) there is no neat bijection: intermediate fields and subgroups need not correspond, and the lattice structures can differ markedly; one instead works with closure operations or with the Galois group of the normal closure.

Boundary

Valid for finite Galois extensions (equivalently finite, separable, normal). Generalizations exist (infinite Galois theory using profinite groups, correspondence for inseparable extensions with care), but the simple finite bijection does not apply to arbitrary extensions or to non-field algebraic structures.

Semantic Tension

This notion is closely related to the more general order-theoretic idea of a Galois connection; the tension lies between the algebraic, bijective Galois correspondence in field theory and the abstract, possibly non-bijective Galois connections in other areas of mathematics.

Synthesis

The Galois correspondence identifies a precise anti-isomorphism between the lattice of intermediate fields of a Galois extension and the lattice of subgroups of its Galois group, allowing translation of structural questions about extensions into group-theoretic terms and back.