Ramification theory of valuations

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 finite 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 σ[w] = [w  σ] (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 σ : LL; 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 mw 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 σ such that σx  x (mod mw) 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.

The reduced ramification index e(w/v) is independent of w and is denoted e(v). Similarly, the relative degree f(w/v) is also independent of w and is denoted f(v).

See also


  • Fröhlich, A.; Taylor, M.J. (1991). Algebraic number theory. Cambridge studies in advanced mathematics. 27. Cambridge University Press. ISBN 0-521-36664-X. Zbl 0744.11001.
  • Zariski, Oscar; Samuel, Pierre (1976) [1960]. Commutative algebra, Volume II. Graduate Texts in Mathematics. 29. New York, Heidelberg: Springer-Verlag. Chapter VI. ISBN 978-0-387-90171-8. Zbl 0322.13001.

This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.