- Ramification
In

mathematics ,**ramification**is a geometric term used for 'branching out', in the way that thesquare root function, forcomplex number s, can be seen to have two "branches" differing in sign. It is also used from the opposite perspective (branches coming together) as when acovering map degenerates at a point of a space, with some collapsing together of the fibers of the mapping.**In complex analysis**In

complex analysis , the basic model can be taken as the "z" $o$ "z"^{"n"}mapping in the complex plane, near "z" = 0. This is the standard local picture inRiemann surface theory, of ramification of order "n". It occurs for example in theRiemann-Hurwitz formula for the effect of mappings on the genus. See alsobranch point .**In algebraic topology**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" $o$ "z"^{"n"}mapping shows this as a local pattern: if we exclude 0, looking at 0 < |z| < 1 say, we have (from thehomotopy point of view) thecircle mapped 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 , andmonodromy ); since "real" codimension two is "complex" codimension one, the local complex example sets the pattern for higher-dimensionalcomplex manifold s. 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 ambientmanifold , 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. Inalgebraic geometry over any field, by analogy, it also happens in algebraic codimension one.**In algebraic number theory**:"See also

splitting of prime ideals in Galois extensions "Ramification in

algebraic number theory means prime numbers factorising into some repeated prime ideal factors. Let "R" be the ring of integers of analgebraic number field "K" and "P" aprime ideal of "R". For each extension field "L" of "K" we can consider theintegral 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-zeronilpotent element - is not a product offinite field s. The analogy with the Riemann surface case was already pointed out byDedekind andHeinrich M. Weber in 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 inGalois module theory.**In local fields**The more detailed analysis of ramification in number fields can be carried out using extensions of the

p-adic number s, because it is a "local" question. In that case a quantitative measure of ramification is defined forGalois extension s, basically by asking how far theGalois group moves 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 morphism in algebraic geometry. It serves to defineétale morphism s.**See also***

Eisenstein polynomial

*Newton polygon

*Puiseux expansion

*Branched covering **External links***

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