The subgroup of the Galois group of an extension consisting of automorphisms that fix a chosen prime (or place) above a base prime; it is the stabilizer (decomposition subgroup) whose structure controls the local behaviour of the extension at that prime and maps onto the Galois group of the residue field with kernel the inertia group.

Principle

Localize the global Galois action at a chosen place by taking the stabilizer of that place; the decomposition group serves as the bridge between the global Galois group and the local Galois or inertia actions, and is determined only up to conjugacy by choice of prime.

Demonstration

In a finite Galois extension, each prime above p has a decomposition group conjugate to any other above p; the decomposition group mod inertia yields the cyclic group generated by Frobenius for unramified primes and in local terms identifies with Gal(K_v^sep/K_v) for completions.

Misapplication

Treating the decomposition group as canonical without noting dependence on the chosen prime or forgetting that statements about decomposition groups are only well‑defined up to conjugacy; or identifying it with the whole Galois group without checking inertia and splitting behaviour.

Consequence

Understanding decomposition groups gives the correct local Galois picture: it locates Frobenius elements, describes ramification via the inertia subgroup, and allows passage between global and local arithmetic such as local class field theory and local factors of representations.

Reversal

Rather than passing to the decomposition group, consider global conjugacy classes or the full Galois group; this emphasizes global symmetry rather than the localized stabilizer and obscures local residue and ramification structure.

Boundary

Defined relative to a chosen prime in an extension and naturally a subgroup determined up to conjugacy; for infinite extensions one considers inverse limits of decomposition groups, and archimedean places require special treatment (complex conjugation phenomena) outside the residue‑field framework.

Semantic Tension

Competes with the view of the local Galois group of a completion: the decomposition group is isomorphic to the absolute Galois group of the completion only after choosing embeddings, and there is tension between canonical global descriptions and the conjugacy dependence of local stabilizers.

Synthesis

The decomposition group is the stabiliser of a place inside the global Galois group; by mapping onto residue‑field Galois groups with inertia kernel it organises the local data (Frobenius, ramification, local Galois action) that mediate between global field extensions and their local completions.