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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 ...
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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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 ))⋯))).