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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 .
A Lubin–Tate extension of a local field K is an abelian extension of K obtained by considering the p-division points of a Lubin–Tate group. If g is an Eisenstein polynomial , f ( t ) = t g ( t ) and F the Lubin–Tate formal group, let θ n denote a root of gf n -1 ( t )= g ( f ( f (⋯( f ( t ))⋯))).
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 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 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 ...
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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 ...
In algebraic geometry, an unramified morphism is a morphism: of schemes such that (a) it is locally of finite presentation and (b) for each and = (), we have that . The residue field is a separable algebraic extension of ().