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In geometry, ramification is 'branching out', in the way that the square root function, for complex numbers, can be seen to have two branches differing in sign. The term is also used from the opposite perspective (branches coming together) as when a covering map degenerates at a point of a space, with some collapsing of the fibers of the mapping.
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. [1] [2] The structure of the set of extensions is known better when L/K is Galois.
The group multiplication is not abelian. The group was studied by number theorists as the group of wild automorphisms of the local field F p ((t)) and by group theorists including D. Johnson (1988) and the name "Nottingham group" refers to his former domicile. This group is a finitely generated pro-p-group, of finite width. For every finite ...
The set of all such extensions is studied in the ramification theory of valuations. Let L/K be a finite extension and let w be an extension of v to L. The index of Γ v in Γ w, e(w/v) = [Γ w : Γ v], is called the reduced ramification index of w over v. It satisfies e(w/v) ≤ [L : K] (the degree of the extension L/K).
More precisely, the conductor computes the non-triviality of higher ramification groups: if s is the largest integer for which the "lower numbering" higher ramification group G s is non-trivial, then (/) = / +, where η L/K is the function that translates from "lower numbering" to "upper numbering" of higher ramification groups.
This is an example of a pure cubic field, and hence of a complex cubic field. In fact, of all pure cubic fields, it has the smallest discriminant (in absolute value), namely −108. [2] The complex cubic field obtained by adjoining to Q a root of x 3 + x 2 − 1 is not pure. It has the smallest discriminant (in absolute value) of all cubic ...
The winding number of () with respect to the point () is a positive integer called the ramification index of . If the ramification index is greater than 1, then z 0 {\displaystyle z_{0}} is called a ramification point of f {\displaystyle f} , and the corresponding critical value f ( z 0 ) {\displaystyle f(z_{0})} is called an (algebraic) branch ...
The tame ramification part ε is defined in terms of the reduction type: ε=0 for good reduction, ε=1 for multiplicative reduction and ε=2 for additive reduction. The wild ramification term δ is zero unless p divides 2 or 3, and in the latter cases it is defined in terms of the wild ramification of the extensions of K by the division points ...