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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.
In 2016, Covanov and Thomé proposed an integer multiplication algorithm based on a generalization of Fermat primes that conjecturally achieves a complexity bound of ( ). This matches the 2015 conditional result of Harvey, van der Hoeven, and Lecerf but uses a different algorithm and relies on a different conjecture. [ 21 ]
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 ...
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 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 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.
An integer that occurs as the discriminant of a quadratic number field is called a fundamental discriminant. [3] Cyclotomic fields: let n > 2 be an integer, let ζ n be a primitive nth root of unity, and let K n = Q(ζ n) be the nth cyclotomic field. The discriminant of K n is given by [2] [4]