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In mathematics, the ramification theory of valuations studies the set of extensions of a valuation v of a field K to an extension L of K. It is a generalization of the ramification theory of Dedekind domains.
Galois case
The structure of the set of extensions is known better when L/K is Galois.
Decomposition group and inertia group
Let (K, v) be a valued field and let L be a Galois extension of K. Let Sv be the set of equivalence classes of extensions of v to L and let G be the Galois group of L over K. Then G acts on Sv by g[w] = [w ○ g] (i.e. w is a representative of the equivalence class [w] ∈ Sv and [w] is sent to the equivalence class of the composition of w with the automorphism g : L → L; this is independent of the choice of w in [w]). In fact, this action is transitive.
Given a fixed extension w of v to L, the decomposition group of w is the stabilizer subgroup Gw of [w], i.e. it is the subgroup of G consisting of all elements that fix the equivalence class [w] ∈ Sv.
Let pw denote the maximal ideal of w inside the valuation ring Rw of w. The inertia group of w is the subgroup Iw of Gw consisting of elements g such that gx ≡ x (mod pw) for all x in Rw. In other words, Iw consists of the elements of the decomposition group that act trivially on the residue field of w. It is a normal subgroup of Gw.
References
- Chapter VI of Zariski, Oscar; Samuel, Pierre (1976) [1960], Commutative algebra, Volume II, Graduate Texts in Mathematics, vol. 29, New York, Heidelberg: Springer-Verlag, ISBN 978-0-387-90171-8
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