# Dictionary Definition

ramify

### Verb

1 have or develop complicating consequences;
"These actions will ramify" [syn: complexify]

2 grow and send out branches or branch-like
structures; "these plants ramify early and get to be very large"
[syn: branch]

3 divide into two or more branches so as to form
a fork; "The road forks" [syn: branch, fork, furcate, separate] [also: ramified]ramified See ramify

# User Contributed Dictionary

## English

### Verb

ramified- past of ramify

# Extensive Definition

In mathematics, ramification is
a geometric term used for 'branching out', in the way that the
square
root function, 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 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 \to zn mapping in the complex plane, near z = 0. This is the standard local picture in Riemann surface theory, of ramification of order n. It occurs for example in the Riemann-Hurwitz formula for the effect of mappings on the genus. See also branch 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 \to zn 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
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 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,
by analogy, it also happens in algebraic codimension one.

## In algebraic number theory

Ramification in algebraic
number theory means 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 ideal
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 element - is
not a product of finite
fields. The analogy with the Riemann surface case was already
pointed out by Dedekind and
Heinrich
M. Weber in the nineteenth century.

The ramification is tame when the e(i) are all
relatively prime to the residue characteristic p of P. This
condition is important in Galois
module theory.

## 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 algebraic geometry

There is also corresponding notion of unramified morphism in algebraic geometry. It serves to define étale morphisms.## See also

## External links

ramified in German: Verzweigung (Algebra)

ramified in French: Ramification

ramified in Finnish:
Haaroittuminen