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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.
Let / be a finite Galois extension of nonarchimedean local fields with finite residue fields / and Galois group.Then the following are equivalent. (i) / is unramified. (ii) / is a field, where is the maximal ideal of .
The conductor of an abelian extension L/K of number fields can be defined, similarly to the local case, using the Artin map. Specifically, let θ : I m → Gal(L/K) be the global Artin map where the modulus m is a defining modulus for L/K; we say that Artin reciprocity holds for m if θ factors through the ray class group modulo m.
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The more detailed analysis of ramification in number fields can be carried out using extensions of the p-adic numbers, because it is a local question. In that case a quantitative measure of ramification is defined for Galois extensions , basically by asking how far the Galois group moves field elements with respect to the metric.
If it is bigger than 1 for some j, the field extension L/K is called ramified at p (or we say that p ramifies in L, or that it is ramified in L). Otherwise, L / K is called unramified at p . If this is the case then by the Chinese remainder theorem the quotient O L / pO L is a product of fields F j .
In mathematics, Abhyankar's lemma (named after Shreeram Shankar Abhyankar) allows one to kill tame ramification by taking an extension of a base field.. More precisely, Abhyankar's lemma states that if A, B, C are local fields such that A and B are finite extensions of C, with ramification indices a and b, and B is tamely ramified over C and b divides a, then the compositum AB is an unramified ...
Rep. Ayanna Pressley will reintroduce H.R. 40, federal legislation to study reparations for slavery, on Wednesday as the Trump administration leads a wide-scale rollback of diversity, equity and ...