View source for Ramification (mathematics)

{{Short description|Branching out of a mathematical structure}}
{{Other uses|Ramification (disambiguation){{!}}Ramification}}
[[Image:Schematic depiction of ramification.svg|right|thumb|300px|Schematic depiction of ramification: the fibers of almost all points in ''Y'' below consist of three points, except for two points in ''Y'' marked with dots, where the fibers consist of one and two points (marked in black), respectively. The map ''f'' is said to be ramified in these points of ''Y''.]]
In [[geometry]], '''ramification''' is 'branching out', in the way that the [[square root]] function, for [[complex number]]s, can be seen to have two ''branches'' differing in sign. The term is also used from the opposite perspective (branches coming together) as when a [[covering map]] [[degeneracy (mathematics)|degenerates]] at a point of a space, with some collapsing of the fibers of the mapping.

==In complex analysis==
{{see also|Branch point}}
[[File:Riemann surface sqrt.svg|thumb|Using the [[Riemann surface]] of the square root]]
In [[complex analysis]], the basic model can be taken as the ''z''&nbsp;&rarr;&nbsp;''z''<sup>''n''</sup> mapping in the complex plane, near&nbsp;''z''&nbsp;=&nbsp;0. This is the standard local picture in [[Riemann surface]] theory, of ramification of order&nbsp;''n''. It occurs for example in the [[Riemann–Hurwitz formula]] for the effect of mappings on the [[genus (mathematics)|genus]].

==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'' &rarr;&nbsp;''z''<sup>''n''</sup> 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 with the whole [[disk (mathematics)|disk]] the Euler–Poincaré characteristic is 1, ''n''&nbsp;–&nbsp;1 being the 'lost' points as the ''n'' sheets come together at&nbsp;''z''&nbsp;=&nbsp;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 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 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 geometry]] over any [[field (mathematics)|field]], by analogy, it also happens in algebraic codimension one.

==In algebraic number theory==
=== In algebraic extensions of the rational numbers ===
{{see also|Splitting of prime ideals in Galois extensions}}

Ramification in [[algebraic number theory]] means a prime ideal factoring in an extension so as to give some repeated prime ideal factors. Namely, let <math>\mathcal{O}_K</math> be the [[ring of integers]] of an [[algebraic number field]] <math>K</math>, and <math>\mathfrak{p}</math> a [[prime ideal]] of <math>\mathcal{O}_K</math>. For a field extension <math>L/K</math> we can consider the
ring of integers <math>\mathcal{O}_L</math> (which is the [[integral closure]] of <math>\mathcal{O}_K</math> in <math>L</math>), and the ideal <math>\mathfrak{p}\mathcal{O}_L</math> of <math>\mathcal{O}_L</math>. This ideal may or may not be prime, but for finite <math>[L:K]</math>, it has a factorization into prime ideals:

:<math>\mathfrak{p}\cdot \mathcal{O}_L = \mathfrak{p}_1^{e_1}\cdots\mathfrak{p}_k^{e_k}</math>

where the <math>\mathfrak{p}_i</math> are distinct prime ideals of <math>\mathcal{O}_L</math>. Then <math>\mathfrak{p}</math> is said to '''ramify''' in <math>L</math> if <math>e_i > 1</math> for some <math>i</math>; otherwise it is '''{{visible anchor|unramified}}'''. In other words, <math>\mathfrak{p}</math> ramifies in <math>L</math> if the '''ramification index''' <math>e_i</math> is greater than one for some <math>\mathfrak{p}_i</math>. An equivalent condition is that <math>\mathcal{O}_L/\mathfrak{p}\mathcal{O}_L</math> has a non-zero [[nilpotent]] element: it is not a product of [[finite field]]s. The analogy with the Riemann surface case was already pointed out by [[Richard Dedekind]] and [[Heinrich M. Weber]] in the nineteenth century.

The ramification is encoded in <math>K</math> by the [[relative discriminant]] and in <math>L</math> by the [[relative different]].  The former is an ideal of <math>\mathcal{O}_K</math> and is divisible  by <math>\mathfrak{p}</math> if and only if some ideal <math>\mathfrak{p}_i</math> of <math>\mathcal{O}_L</math> dividing <math>\mathfrak{p}</math> is ramified.  The latter is an ideal of <math>\mathcal{O}_L</math> and is divisible by the prime ideal <math>\mathfrak{p}_i</math> of <math>\mathcal{O}_L</math> precisely when <math>\mathfrak{p}_i</math> is ramified.

The ramification is '''tame''' when the ramification indices <math>e_i</math> are all relatively prime to the residue characteristic ''p'' of <math>\mathfrak{p}</math>, otherwise '''wild'''. This condition is important in [[Galois module]] theory. A finite generically étale extension <math>B/A</math> of [[Dedekind domain]]s is tame if and only if the trace <math>\operatorname{Tr}: B \to A</math> is surjective.

===In local fields===
{{main|Ramification of 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 for [[Galois extension]]s, basically by asking how far the [[Galois group]] moves field elements with respect to the metric. A sequence of [[ramification group]]s is defined, reifying (amongst other things) ''wild'' (non-tame) ramification. This goes beyond the geometric analogue.

==In algebra==
{{main|Ramification theory of valuations}}
In [[valuation theory]], the [[ramification theory of valuations]] studies the set of [[extension of a valuation|extensions]] of a [[valuation (algebra)|valuation]] of a [[field (mathematics)|field]] ''K'' to an [[extension field]] of ''K''. This generalizes the notions in algebraic number theory, local fields, and Dedekind domains.

==In algebraic geometry==
There is also corresponding notion of [[unramified morphism]] in algebraic geometry. It serves to define [[étale morphism]]s.

Let <math>f: X \to Y</math> be a morphism of schemes. The support of the quasicoherent sheaf <math>\Omega_{X/Y}</math> is called the '''ramification locus''' of <math>f</math> and the image of the ramification locus, <math>f\left( \operatorname{Supp} \Omega_{X/Y} \right)</math>, is called the '''branch locus''' of <math>f</math>. If <math>\Omega_{X/Y}=0</math> we say that <math>f</math> is '''formally unramified''' and if <math>f</math> is also of locally finite presentation we say that <math>f</math> is '''[[unramified morphism|unramified]]''' (see {{harvnb|Vakil|2017}}).

== See also ==
* [[Eisenstein polynomial]]
* [[Newton polygon]]
* [[Puiseux expansion]]
* [[Branched covering]]
{{wiktionary}}

==References==
* {{Neukirch ANT}}
* {{cite book|last=Vakil|first=Ravi|title=The Rising Sea: Foundations of algebraic geometry|url=http://math.stanford.edu/~vakil/216blog/FOAGnov1817public.pdf|date=18 November 2017|access-date=5 June 2019}}

== External links ==
* {{planetmath_reference|urlname=SplittingAndRamificationInNumberFieldsAndGaloisExtensions|title=Splitting and ramification in number fields and Galois extensions}}

[[Category:Algebraic number theory]]
[[Category:Algebraic topology]]
[[Category:Complex analysis]]