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The ramification is tame when the ramification indices are all relatively prime to the residue characteristic p of , otherwise wild. This condition is important in Galois module theory. A finite generically étale extension B / A {\displaystyle B/A} of Dedekind domains is tame if and only if the trace Tr : B → A {\displaystyle \operatorname ...
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.
Graphs of functions commonly used in the analysis of algorithms, showing the number of operations versus input size for each function. The following tables list the computational complexity of various algorithms for common mathematical operations.
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 following procedure (Neukirch, p. 47) solves this problem in many cases. The strategy is to select an integer θ in O L so that L is generated over K by θ (such a θ is guaranteed to exist by the primitive element theorem), and then to examine the minimal polynomial H(X) of θ over K; it is a monic polynomial with coefficients in O K.
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).
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 ...