Ramification (mathematics)

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

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 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

P₁^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 some e(i) > 1. 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 Weber in the nineteenth century.

The ramification is tame when the e(i) are all less than 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.