The decomposition group of a prime ideal 
in a finite Galois extension 
is the subgroup of the Galois
group consisting of automorphisms that fix 
. Its action on the residue field has kernel the inertia group,
and the quotient is the Galois group of the residue-field
extension.
Decomposition Group
See also
Galois Group, Ideal, Prime IdealExplore with Wolfram|Alpha
References
Neukirch, J. Algebraic Number Theory. Berlin, Germany: Springer-Verlag, 1999.Cite this as:
Weisstein, Eric W. "Decomposition Group." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/DecompositionGroup.html