mathematics, ramification is a geometric term used for 'branching out', in the way that the square rootfunction, for complex numbers, can be seen to have two "branches" differing in sign. It is also used from the opposite perspective (branches coming together) as when a covering mapdegenerates at a point of a space, with some collapsing together of the fibers of the mapping.
In complex analysis
complex analysis, the basic model can be taken as the "z" "z""n" mapping in the complex plane, near "z" = 0. This is the standard local picture in Riemann surfacetheory, of ramification of order "n". It occurs for example in the Riemann-Hurwitz formulafor the effect of mappings on the genus. See also branch point.
In algebraic topology
In a covering map the
Euler-Poincaré characteristicshould 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 homotopypoint of view) the circlemapped to itself by the "n"-th power map (Euler-Poincaré characteristic 0), but with the whole disk the Euler-Poincaré characteristic is 1, "n"-1 being the 'lost' points as the "n" sheets come together at "z" = 0.
In geometric terms, ramification is something that happens in "codimension two" (like
knot theory, and monodromy); since "real" codimension two is "complex" codimension one, the local complex example sets the pattern for higher-dimensional complex manifolds. In complex analysis, sheets can't simply fold over along a line (one variable), or codimension one subspace in the general case. The ramification set (branch locus on the base, double point set above) will be two real dimensions lower than the ambient manifold, and so will not separate it into two 'sides', locally - there will be paths that trace round the branch locus, just as in the example. In algebraic geometryover any field, by analogy, it also happens in algebraic codimension one.
In algebraic number theory
splitting of prime ideals in Galois extensions"
algebraic number theorymeans prime numbers factorising into some repeated prime ideal factors. Let "R" be the ring of integers of an algebraic number field"K" and "P" a prime idealof "R". For each extension field "L" of "K" we can consider the integral closure"S" of "R" in "L" and the ideal "PS" of "S". This may or may not be prime, but assuming ["L":"K"] is finite it is a product of prime ideals
:"P"1"e"(1) ... "P""k""e"("k")
where the "P""i" are distinct prime ideals of "S". Then "P" is said to ramify in "L" if "e"("i") > 1 for some "i". In other words, "P" ramifies in "L" if the ramification index "e"("i") is greater than one for any "P""i". An equivalent condition is that "S"/"PS" has a non-zero
nilpotentelement - is not a product of finite fields. The analogy with the Riemann surface case was already pointed out by Dedekindand Heinrich M. Weberin the nineteenth century.
The ramification is tame when the ramification indices "e"("i") are all relatively prime to the residue characteristic "p" of P, otherwise wild. This condition is important in
In local fields
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 groupmoves field elements with respect to the metric. A sequence of ramification groups is defined, reifying (amongst other things) "wild" (non-tame) ramification. This goes beyond the geometric analogue.
In algebraic geometry
There is also corresponding notion of
unramified morphismin algebraic geometry. It serves to define étale morphisms.
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