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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 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.
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.
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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.
The splitting of prime ideals in Galois extensions is sometimes attributed to David Hilbert by calling it Hilbert theory. There is a geometric analogue, for ramified coverings of Riemann surfaces, which is simpler in that only one kind of subgroup of G need be considered, rather than two. This was certainly familiar before Hilbert.
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 ...
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