The group of field automorphisms of a field extension E over a base field F that fix F pointwise; it encodes algebraic symmetries of the extension and of polynomial roots in E.

Principle

Automorphisms that preserve the base field compose to form a finite or profinite group whose subgroup structure corresponds (under the Galois correspondence) to intermediate field extensions and algebraic relations among roots.

Demonstration

For the splitting field of x^3 − 2 over Q the Galois group is isomorphic to S3: it permutes the three roots while respecting field operations and the rational numbers fixed by every automorphism.

Misapplication

Confusing the Galois group with the full permutation group of a polynomial's formal roots without regard for field relations or treating coefficient permutations as automorphisms can overcount symmetries that are not field automorphisms.

Consequence

Knowledge of the Galois group determines properties such as solvability by radicals, degrees of intermediate extensions, and ramification behavior in number fields; group-theoretic features translate into algebraic field properties.

Reversal

Considering ring automorphisms or topological symmetry groups of geometric objects in the absence of a field structure inverts the context: those groups do not generally control algebraic solvability of polynomials.

Boundary

Defined only for field extensions (and generalized to Galois categories/profinite groups); it excludes arbitrary ring extensions and maps that do not preserve multiplicative inverses or the base field pointwise.

Semantic Tension

Sometimes conflated with the mere permutations of roots: the Galois group is the subgroup of root permutations realized by field automorphisms compatible with algebraic relations and base-field fixation.

Synthesis

The Galois group is the symmetry group of a field extension composed of automorphisms fixing the base field, and via the fundamental Galois correspondence its subgroup lattice mirrors the lattice of intermediate fields and algebraic dependencies.