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In a covering map the Euler–Poincaré characteristic should multiply by the number of sheets; ramification can therefore be detected by some dropping from that. The z → z n mapping shows this as a local pattern: if we exclude 0, looking at 0 < |z| < 1 say, we have (from the homotopy point of view) the circle mapped to itself by the n-th power map (Euler–Poincaré characteristic 0), but ...
= / is tamely ramified (i.e., the ramification index is prime to the residue characteristic.) The study of ramification groups reduces to the ...
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 ...
[11] [16] The precise exponent to which a ramified prime P divides δ is termed the differential exponent of P and is equal to e − 1 if P is tamely ramified: that is, when P does not divide e. [17] In the case when P is wildly ramified the differential exponent lies in the range e to e + eν P (e) − 1.
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 tame fundamental group of some scheme U is a quotient of the usual fundamental group of which takes into account only covers that are tamely ramified along , where is some compactification and is the complement of in .
There, given a Galois ramified cover, all but finitely many points have the same number of preimages. The splitting of primes in extensions that are not Galois may be studied by using a splitting field initially, i.e. a Galois extension that is somewhat larger. For example, cubic fields usually are 'regulated' by a degree 6 field containing them.
The technical assumption needed for the Moy–Prasad isomorphism to exist is that be tame, meaning that splits over a tamely ramified extension of the base field . If this assumption is violated then g x , r : s {\displaystyle {\mathfrak {g}}_{x,r:s}} and G ( k ) x , r : s {\displaystyle G(k)_{x,r:s}} are not necessarily isomorphic.