Search results
Results From The WOW.Com Content Network
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.
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 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.
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.
Integer multiplication respects the congruence classes, that is, a ≡ a' and b ≡ b' (mod n) implies ab ≡ a'b' (mod n). This implies that the multiplication is associative, commutative, and that the class of 1 is the unique multiplicative identity. Finally, given a, the multiplicative inverse of a modulo n is an integer x satisfying ax ≡ ...
In mathematics, the Néron–Ogg–Shafarevich criterion states that if A is an elliptic curve or abelian variety over a local field K and ℓ is a prime not dividing the characteristic of the residue field of K then A has good reduction if and only if the ℓ-adic Tate module T ℓ of A is unramified.
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).