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In
geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
, ramification is 'branching out', in the way that the
square root In mathematics, a square root of a number is a number such that ; in other words, a number whose '' square'' (the result of multiplying the number by itself, or  ⋅ ) is . For example, 4 and −4 are square roots of 16, because . ...
function, for
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...
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
degenerates Degenerates is a musical group which originated in Grosse Pointe Park, Michigan in 1979, during the formative years of the Detroit hardcore scene. The group predated the Process of Elimination EP, which some reviewers view as the beginning of the ...
at a point of a space, with some collapsing of the fibers of the mapping.


In complex analysis

In
complex analysis Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates Function (mathematics), functions of complex numbers. It is helpful in many branches of mathemati ...
, 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 surface In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed ver ...
theory, of ramification of order ''n''. It occurs for example in the
Riemann–Hurwitz formula In mathematics, the Riemann–Hurwitz formula, named after Bernhard Riemann and Adolf Hurwitz, describes the relationship of the Euler characteristics of two surfaces when one is a ''ramified covering'' of the other. It therefore connects ramifi ...
for the effect of mappings on the
genus Genus ( plural genera ) is a taxonomic rank used in the biological classification of living and fossil organisms as well as viruses. In the hierarchy of biological classification, genus comes above species and below family. In binomial nom ...
. See also
branch point In the mathematical field of complex analysis, a branch point of a multi-valued function (usually referred to as a "multifunction" in the context of complex analysis) is a point such that if the function is n-valued (has n values) at that 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'' → ''z''''n'' mapping shows this as a local pattern: if we exclude 0, looking at 0 < , ''z'', < 1 say, we have (from the
homotopy In topology, a branch of mathematics, two continuous functions from one topological space to another are called homotopic (from grc, ὁμός "same, similar" and "place") if one can be "continuously deformed" into the other, such a deform ...
point of view) the
circle A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is cons ...
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 In the mathematical field of topology, knot theory is the study of mathematical knots. While inspired by knots which appear in daily life, such as those in shoelaces and rope, a mathematical knot differs in that the ends are joined so it cannot ...
, 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 In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
, 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 Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
over any field, by analogy, it also happens in algebraic codimension one.


In algebraic number theory


In algebraic extensions of \mathbb

Ramification in
algebraic number theory Algebraic number theory is a branch of number theory that uses the techniques of abstract algebra to study the integers, rational numbers, and their generalizations. Number-theoretic questions are expressed in terms of properties of algebraic o ...
means a prime ideal factoring in an extension so as to give some repeated prime ideal factors. Namely, let \mathcal_K be the ring of integers of an algebraic number field K, and \mathfrak a
prime ideal In algebra, a prime ideal is a subset of a ring that shares many important properties of a prime number in the ring of integers. The prime ideals for the integers are the sets that contain all the multiples of a given prime number, together wi ...
of \mathcal_K. For a field extension L/K we can consider the ring of integers \mathcal_L (which is the integral closure of \mathcal_K in L), and the ideal \mathfrak\mathcal_L of \mathcal_L. This ideal may or may not be prime, but for finite :K/math>, it has a factorization into prime ideals: :\mathfrak\cdot \mathcal_L = \mathfrak_1^\cdots\mathfrak_k^ where the \mathfrak_i are distinct prime ideals of \mathcal_L. Then \mathfrak is said to ramify in L if e_i > 1 for some i; otherwise it is . In other words, \mathfrak ramifies in L if the ramification index e_i is greater than one for some \mathfrak_i. An equivalent condition is that \mathcal_L/\mathfrak\mathcal_L has a non-zero
nilpotent In mathematics, an element x of a ring R is called nilpotent if there exists some positive integer n, called the index (or sometimes the degree), such that x^n=0. The term was introduced by Benjamin Peirce in the context of his work on the cl ...
element: it is not a product of
finite field In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtr ...
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 K by the
relative discriminant In mathematics, the discriminant of an algebraic number field is a numerical invariant that, loosely speaking, measures the size of the (ring of integers of the) algebraic number field. More specifically, it is proportional to the squared volum ...
and in L by the relative different. The former is an ideal of \mathcal_K and is divisible by \mathfrak if and only if some ideal \mathfrak_i of \mathcal_L dividing \mathfrak is ramified. The latter is an ideal of \mathcal_L and is divisible by the prime ideal \mathfrak_i of \mathcal_L precisely when \mathfrak_i is ramified. The ramification is tame when the ramification indices e_i are all relatively prime to the residue characteristic ''p'' of \mathfrak, otherwise wild. This condition is important in Galois module theory. A finite generically étale extension B/A of Dedekind domains is tame if and only if the trace \operatorname: B \to A is surjective.


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 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 algebra

In
valuation theory In algebra (in particular in algebraic geometry or algebraic number theory), a valuation is a function on a field that provides a measure of size or multiplicity of elements of the field. It generalizes to commutative algebra the notion of size inhe ...
, the ramification theory of valuations studies the set of
extensions Extension, extend or extended may refer to: Mathematics Logic or set theory * Axiom of extensionality * Extensible cardinal * Extension (model theory) * Extension (predicate logic), the set of tuples of values that satisfy the predicate * Ex ...
of a valuation of a 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, an unramified morphism is a morphism f: X \to Y of schemes such that (a) it is locally of finite presentation and (b) for each x \in X and y = f(x), we have that # The residue field k(x) is a separable algebraic extension of ...
in algebraic geometry. It serves to define étale morphisms. Let f: X \to Y be a morphism of schemes. The support of the quasicoherent sheaf \Omega_ is called the ramification locus of f and the image of the ramification locus, f\left( \operatorname \Omega_ \right), is called the branch locus of f. If \Omega_=0 we say that f is formally unramified and if f is also of locally finite presentation we say that f is
unramified In geometry, ramification is 'branching out', in the way that the square root function, for complex numbers, can be seen to have two ''branches'' differing in sign. The term is also used from the opposite perspective (branches coming together) as ...
(see ).


See also

* Eisenstein polynomial * Newton polygon * Puiseux expansion * Branched covering


References

* *


External links

* {{planetmath_reference, urlname=SplittingAndRamificationInNumberFieldsAndGaloisExtensions, title=Splitting and ramification in number fields and Galois extensions Algebraic number theory Algebraic topology Complex analysis