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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 ...
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.
[1] [2] [3] Examples of wild problems are classifying indecomposable representations of any quiver that is neither a Dynkin quiver (i.e. the underlying undirected graph of the quiver is a (finite) Dynkin diagram) nor a Euclidean quiver (i.e., the underlying undirected graph of the quiver is an affine Dynkin diagram). [4]
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.
Many mathematical problems have been stated but not yet solved. These problems come from many areas of mathematics, such as theoretical physics, computer science, algebra, analysis, combinatorics, algebraic, differential, discrete and Euclidean geometries, graph theory, group theory, model theory, number theory, set theory, Ramsey theory, dynamical systems, and partial differential equations.
The Clay Mathematics Institute officially designated the title Millennium Problem for the seven unsolved mathematical problems, the Birch and Swinnerton-Dyer conjecture, Hodge conjecture, Navier–Stokes existence and smoothness, P versus NP problem, Riemann hypothesis, Yang–Mills existence and mass gap, and the Poincaré conjecture at the ...
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 ...
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).