Ramification

# Ramification

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Encyclopedia

In mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, ramification is a geometric term used for 'branching out', in the way that the square root
Square root
In mathematics, a square root of a number x is a number r such that r2 = x, or, in other words, a number r whose square is x...

function, for complex number
Complex number
A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...

s, 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 map
Covering map
In mathematics, more specifically algebraic topology, a covering map is a continuous surjective function p from a topological space, C, to a topological space, X, such that each point in X has a neighbourhood evenly covered by p...

degenerates
Degeneracy (mathematics)
In mathematics, a degenerate case is a limiting case in which a class of object changes its nature so as to belong to another, usually simpler, class....

at a point of a space, with some collapsing together of the fibers of the mapping.

## In complex analysis

In complex analysis
Complex analysis
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is useful in many branches of mathematics, including number theory and applied mathematics; as well as in physics,...

, the basic model can be taken as the z zn mapping in the complex plane, near z = 0. This is the standard local picture in Riemann surface
Riemann surface
In mathematics, particularly in complex analysis, a Riemann surface, first studied by and named after Bernhard Riemann, is a one-dimensional complex manifold. Riemann surfaces can be thought of as "deformed versions" of the complex plane: locally near every point they look like patches of the...

theory, of ramification of order n. It occurs for example in the Riemann–Hurwitz formula for the effect of mappings on the genus
Genus (mathematics)
In mathematics, genus has a few different, but closely related, meanings:-Orientable surface:The genus of a connected, orientable surface is an integer representing the maximum number of cuttings along non-intersecting closed simple curves without rendering the resultant manifold disconnected. It...

Branch point
In the mathematical field of complex analysis, a branch point of a multi-valued function is a point such that the function is discontinuous when going around an arbitrarily small circuit around this 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 zn mapping shows this as a local pattern: if we exclude 0, looking at 0 < |z| < 1 say, we have (from the homotopy
Homotopy
In topology, two continuous functions from one topological space to another are called homotopic if one can be "continuously deformed" into the other, such a deformation being called a homotopy between the two functions...

point of view) the circle
Circle
A circle is a simple shape of Euclidean geometry consisting of those points in a plane that are a given distance from a given point, the centre. The distance between any of the points and the centre is called the radius....

mapped to itself by the n-th power map (Euler-Poincaré characteristic 0), but with the whole disk
Disk (mathematics)
In geometry, a disk is the region in a plane bounded by a circle.A disk is said to be closed or open according to whether or not it contains the circle that constitutes its boundary...

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
Knot theory
In topology, knot theory is the study of mathematical knots. While inspired by knots which appear in daily life in shoelaces and rope, a mathematician's knot differs in that the ends are joined together so that it cannot be undone. In precise mathematical language, a knot is an embedding of a...

, and monodromy
Monodromy
In mathematics, monodromy is the study of how objects from mathematical analysis, algebraic topology and algebraic and differential geometry behave as they 'run round' a singularity. As the name implies, the fundamental meaning of monodromy comes from 'running round singly'...

); since real codimension two is complex codimension one, the local complex example sets the pattern for higher-dimensional complex manifold
Complex manifold
In differential geometry, a complex manifold is a manifold with an atlas of charts to the open unit disk in Cn, such that the transition maps are holomorphic....

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
Manifold
In mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the 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
Algebraic geometry
Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...

over any field
Field (mathematics)
In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...

, by analogy, it also happens in algebraic codimension one.

### In algebraic extensions of

Splitting of prime ideals in Galois extensions
In mathematics, the interplay between the Galois group G of a Galois extension L of a number field K, and the way the prime ideals P of the ring of integers OK factorise as products of prime ideals of OL, provides one of the richest parts of algebraic number theory...

Ramification in algebraic number theory
Algebraic number theory
Algebraic number theory is a major branch of number theory which studies algebraic structures related to algebraic integers. This is generally accomplished by considering a ring of algebraic integers O in an algebraic number field K/Q, and studying their algebraic properties such as factorization,...

means prime numbers factorising into some repeated prime ideal factors. Let R be the ring of integers
Ring of integers
In mathematics, the ring of integers is the set of integers making an algebraic structure Z with the operations of integer addition, negation, and multiplication...

of an algebraic number field
Algebraic number field
In mathematics, an algebraic number field F is a finite field extension of the field of rational numbers Q...

K and P a prime ideal
Prime ideal
In algebra , a prime ideal is a subset of a ring which shares many important properties of a prime number in the ring of integers...

of 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
P1e(1) ... Pke(k)

where the Pi 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 Pi. An equivalent condition is that S/PS has a non-zero nilpotent
Nilpotent
In mathematics, an element x of a ring R is called nilpotent if there exists some positive integer n such that xn = 0....

element: it is not a product of finite field
Finite field
In abstract algebra, a finite field or Galois field is a field that contains a finite number of elements. Finite fields are important in number theory, algebraic geometry, Galois theory, cryptography, and coding theory...

s. The analogy with the Riemann surface case was already pointed out by Richard Dedekind
Richard Dedekind
Julius Wilhelm Richard Dedekind was a German mathematician who did important work in abstract algebra , algebraic number theory and the foundations of the real numbers.-Life:...

and Heinrich 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 in Galois module
Galois module
In mathematics, a Galois module is a G-module where G is the Galois group of some extension of fields. The term Galois representation is frequently used when the G-module is a vector space over a field or a free module over a ring, but can also be used as a synonym for G-module...

theory. A finite generically étale extension of Dedekind domain
Dedekind domain
In abstract algebra, a Dedekind domain or Dedekind ring, named after Richard Dedekind, is an integral domain in which every nonzero proper ideal factors into a product of prime ideals. It can be shown that such a factorization is then necessarily unique up to the order of the factors...

s is tame iff the trace is surjective.

### In local fields

The more detailed analysis of ramification in number fields can be carried out using extensions of the p-adic number
In mathematics, and chiefly number theory, the p-adic number system for any prime number p extends the ordinary arithmetic of the rational numbers in a way different from the extension of the rational number system to the real and complex number systems...

s, because it is a local question. In that case a quantitative measure of ramification is defined for Galois extension
Galois extension
In mathematics, a Galois extension is an algebraic field extension E/F satisfying certain conditions ; one also says that the extension is Galois. The significance of being a Galois extension is that the extension has a Galois group and obeys the fundamental theorem of Galois theory.The definition...

s, basically by asking how far the Galois group
Galois group
In mathematics, more specifically in the area of modern algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension...

moves field elements with respect to the metric. A sequence of ramification group
Ramification group
In number theory, more specifically in local class field theory, the ramification groups are a filtration of the Galois group of a local field extension, which gives a precisely information on the ramification phenomenon of the extension....

s is defined, reifying (amongst other things) wild (non-tame) ramification. This goes beyond the geometric analogue.

## In algebra

In valuation theory, the ramification theory of valuations
Ramification theory of valuations
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...

studies the set of extensions of a valuation of a field
Field (mathematics)
In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...

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
Étale morphism
In algebraic geometry, a field of mathematics, an étale morphism is an algebraic analogue of the notion of a local isomorphism in the complex analytic topology. They satisfy the hypotheses of the implicit function theorem, but because open sets in the Zariski topology are so large, they are not...

s.