ramification index

1 Ramification in number fields

Definition 1 (First definition).

Let L/K be an extensionPlanetmathPlanetmathPlanetmathPlanetmath of number fieldsMathworldPlanetmath. Let 𝔭 be a nonzero prime idealMathworldPlanetmathPlanetmathPlanetmath in the ring of integersMathworldPlanetmath 𝒪K of K, and suppose the ideal 𝔭𝒪L𝒪L factors as


for some prime ideals 𝔓i𝒪L and exponentsPlanetmathPlanetmath ei. The natural numberMathworldPlanetmath ei is called the ramification index of 𝔓i over 𝔭. It is often denoted e(𝔓i/𝔭). If ei>1 for any i, then we say the ideal 𝔭 ramifies in L.

Likewise, if 𝔓 is a nonzero prime ideal in 𝒪L, and 𝔭:=𝔓𝒪K, then we say 𝔓 ramifies over K if the ramification index e(𝔓/𝔭) of 𝔓 in the factorization of the ideal 𝔭𝒪L𝒪L is greater than 1. That is, a prime 𝔭 in 𝒪K ramifies in L if at least one prime 𝔓 dividing 𝔭𝒪L ramifies over K. If L/K is a Galois extensionMathworldPlanetmath, then the ramification indices of all the primes dividing 𝔭𝒪L are equal, since the Galois groupMathworldPlanetmath is transitiveMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath on this set of primes.

1.1 The local view

The phenomenon of ramification has an equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath interpretationMathworldPlanetmathPlanetmath in terms of local ringsMathworldPlanetmath. With L/K as before, let 𝔓 be a prime in 𝒪L with 𝔭:=𝔓𝒪K. Then the induced map of localizationsMathworldPlanetmath (𝒪K)𝔭(𝒪L)𝔓 is a local homomorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath of local rings (in fact, of discrete valuation rings), and the ramification index of 𝔓 over 𝔭 is the unique natural number e such that


An astute reader may notice that this formulation of ramification index does not require that L and K be number fields, or even that they play any role at all. We take advantage of this fact here to give a second, more general definition.

Definition 2 (Second definition).

Let ι:AB be any ring homomorphism. Suppose 𝔓B is a prime ideal such that the localization B𝔓 of B at 𝔓 is a discrete valuation ring. Let 𝔭 be the prime ideal ι-1(𝔓)A, so that ι induces a local homomorphism ι𝔓:A𝔭B𝔓. Then the ramification index e(𝔓/𝔭) is defined to be the unique natural number such that


or if ι(𝔭)B𝔓=(0).

The reader who is not interested in local rings may assume that A and B are unique factorization domainsMathworldPlanetmath, in which case e(𝔓/𝔭) is the exponent of 𝔓 in the factorization of the ideal ι(𝔭)B, just as in our first definition (but without the requirement that the rings A and B originate from number fields).

There is of course much more that can be said about ramification indices even in this purely algebraicMathworldPlanetmathPlanetmath setting, but we limit ourselves to the following remarks:

  1. 1.

    Suppose A and B are themselves discrete valuation rings, with respective maximal idealsMathworldPlanetmath 𝔭 and 𝔓. Let A^:=limA/𝔭n and B^:=limB/𝔓n be the completions of A and B with respect to 𝔭 and 𝔓. Then

    e(𝔓/𝔭)=e(𝔓B^/𝔭A^). (1)

    In other words, the ramification index of 𝔓 over 𝔭 in the AalgebraMathworldPlanetmathPlanetmath B equals the ramification index in the completions of A and B with respect to 𝔭 and 𝔓.

  2. 2.

    Suppose A and B are Dedekind domainsMathworldPlanetmath, with respective fraction fields K and L. If B equals the integral closure of A in L, then

    𝔓𝔭e(𝔓/𝔭)f(𝔓/𝔭)[L:K], (2)

    where 𝔓 ranges over all prime ideals in B that divide 𝔭B, and f(𝔓/𝔭):=dimA/𝔭(B/𝔓) is the inertial degree of 𝔓 over 𝔭. Equality holds in Equation (2) whenever B is finitely generatedMathworldPlanetmathPlanetmathPlanetmath as an A–module.

2 Ramification in algebraic geometry

The word “ramify” in English means “to divide into two or more branches,” and we will show in this sectionPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath that the mathematical term lives up to its common English meaning.

Definition 3 (Algebraic version).

Let f:C1C2 be a non–constant regular morphism of curves (by which we mean one dimensional nonsingularPlanetmathPlanetmath irreduciblePlanetmathPlanetmathPlanetmathPlanetmath algebraic varieties) over an algebraically closed field k. Then f has a nonzero degree n:=degf, which can be defined in any of the following ways:

  • The number of points in a genericPlanetmathPlanetmathPlanetmath fiber f-1(p), for pC2

  • The maximum number of points in f-1(p), for pC2

  • The degree of the extension k(C1)/f*k(C2) of function fieldsMathworldPlanetmath

There is a finite setMathworldPlanetmath of points pC2 for which the inverse imagePlanetmathPlanetmath f-1(p) does not have size n, and we call these points the branch points or ramification points of f. If PC1 with f(P)=p, then the ramification index e(P/p) of f at P is the ramification index obtained algebraically from Definition 2 by taking

  • A=k[C2]p, the local ring consisting of all rational functions in the function field k(C2) which are regularPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath at p.

  • B=k[C1]P, the local ring consisting of all rational functions in the function field k(C1) which are regular at P.

  • 𝔭=𝔪p, the maximal ideal in A consisting of all functions which vanish at p.

  • 𝔓=𝔪P, the maximal ideal in B consisting of all functions which vanish at P.

  • ι=fp*:k[C2]pk[C1]P, the map on the function fields induced by the morphismMathworldPlanetmath f.

Example 4.

The picture in Figure 1 may be worth a thousand words. Let k= and C1=C2==𝔸1. Take the map f: given by f(y)=y2. Then f is plainly a map of degree 2, and every point in C2 except for 0 has two preimages in C1. The point 0 is thus a ramification point of f of index 2, and we have drawn the graph of f near 0.

Figure 1: The function f(y)=y2 near y=0.

Note that we have only drawn the real locus of f because that is all that can fit into two dimensionsMathworldPlanetmathPlanetmath. We see from the figure that a typical point on C2 such as the point x=1 has two points in C1 which map to it, but that the point x=0 has only one corresponding point of C1 which “branches” or “ramifies” into two distinct points of C1 whenever one moves away from 0.

2.1 Relation to the number field case

The relationship between Definition 2 and Definition 3 is easiest to explain in the case where f is a map between affine varietiesMathworldPlanetmath. When C1 and C2 are affine, then their coordinate rings k[C1] and k[C2] are Dedekind domains, and the points of the curve C1 (respectively, C2) correspond naturally with the maximal ideals of the ring k[C1] (respectively, k[C2]). The ramification points of the curve C1 are then exactly the points of C1 which correspond to maximal ideals of k[C1] that ramify in the algebraic sense, with respect to the map f*:k[C2]k[C1] of coordinate rings.

Equation (2) in this case says


and we see that the well known formulaMathworldPlanetmath (2) in number theoryMathworldPlanetmath is simply the algebraic analogue of the geometric fact that the number of points in the fiber of f, counting multiplicitiesMathworldPlanetmath, is always n.

Example 5.

Let f: be given by f(y)=y2 as in Example 4. Since C2 is just the affine line, the coordinate ring [C2] is equal to [X], the polynomial ringMathworldPlanetmath in one variable over . Likewise, [C1]=[Y], and the induced map f*:[X][Y] is naturally given by f*(X)=Y2. We may accordingly identify the coordinate ring [C2] with the subring [X2] of [X]=[C1].

Now, the ring [X] is a principal ideal domainMathworldPlanetmath, and the maximal ideals in [X] are exactly the principal idealsMathworldPlanetmathPlanetmathPlanetmath of the form (X-a) for any a. Hence the nonzero prime ideals in [X2] are of the form (X2-a), and these factor in [X] as


Note that the two prime ideals (X-a) and (X+a) of [X] are equal only when a=0, so we see that the ideal (X2-a) in [X2], corresponding to the point aC2, ramifies in C1 exactly when a=0. We have therefore recovered our previous geometric characterization of the ramified points of f, solely in terms of the algebraic factorizations of ideals in [X].

In the case where f is a map between projective varieties, Definition 2 does not directly apply to the coordinate rings of C1 and C2, but only to those of open covers of C1 and C2 by affine varieties. Thus we do have an instance of yet another new phenomenon here, and rather than keep the reader in suspense we jump straight to the final, most general definition of ramification that we will give.

Definition 6 (Final form).

Let f:(X,𝒪X)(Y,𝒪Y) be a morphism of locally ringed spaces. Let pX and suppose that the stalk (𝒪X)p is a discrete valuation ring. Write ϕp:(𝒪Y)f(p)(𝒪X)p for the induced map of f on stalks at p. Then the ramification index of p over Y is the unique natural number e, if it exists (or if it does not exist), such that


where 𝔪p and 𝔪f(p) are the respective maximal ideals of (𝒪X)p and (𝒪Y)f(p). We say p is ramified in Y if e>1.

Example 7.

A ring homomorphism ι:AB corresponds functorially to a morphism Spec(B)Spec(A) of locally ringed spaces from the prime spectrum of B to that of A, and the algebraic notion of ramification from Definition 2 equals the sheaf–theoretic notion of ramification from Definition 6.

Example 8.

For any morphism of varieties f:C1C2, there is an induced morphism f# on the structure sheaves of C1 and C2, which are locally ringed spaces. If C1 and C2 are curves, then the stalks are one dimensional regular local ringsMathworldPlanetmath and therefore discrete valuation rings, so in this way we recover the algebraic geometric definition (Definition 3) from the sheaf definition (Definition 6).

3 Ramification in complex analysis

Ramification points or branch points in complex geometry are merely a special case of the high–flown terminology of Definition 6. However, they are important enough to merit a separate mention here.

Definition 9 (Analytic version).

Let f:MN be a holomorphic map of Riemann surfacesMathworldPlanetmath. For any pM, there exists local coordinate charts U and V around p and f(p) such that f is locally the map zze from U to V. The natural number e is called the ramification index of f at p, and p is said to be a branch point or ramification point of f if e>1.

Example 10.

Take the map f:, f(y)=y2 of Example 4. We study the behavior of f near the unramified point y=1 and near the ramified point y=0. Near y=1, take the coordinate w=y-1 on the domain and v=x-1 on the range. Then f maps w+1 to (w+1)2, which in the v coordinate is (w+1)2-1=2w+w2. If we change coordinates to z=2w+w2 on the domain, keeping v on the range, then f(z)=z, so the ramification index of f at y=1 is equal to 1.

Near y=0, the function f(y)=y2 is already in the form zze with e=2, so the ramification index of f at y=0 is equal to 2.

3.1 Algebraic–analytic correspondence

Of course, the analyticPlanetmathPlanetmath notion of ramification given in Definition 9 can be couched in terms of locally ringed spaces as well. Any Riemann surface together with its sheaf of holomorphic functionsMathworldPlanetmath is a locally ringed space. Furthermore the stalk at any point is always a discrete valuation ring, because germs of holomorphic functions have Taylor expansionsMathworldPlanetmath making the stalk isomorphic to the power seriesMathworldPlanetmath ring [[z]]. We can therefore apply Definition 6 to any holomorphic map of Riemann surfaces, and it is not surprising that this process yields the same results as Definition 9.

More generally, every map of algebraic varieties f:VW can be interpreted as a holomorphic map of Riemann surfaces in the usual way, and the ramification points on V and W under f as algebraic varieties are identical to their ramification points as Riemann surfaces. It turns out that the analytic structure may be regarded in a certain sense as the “completion” of the algebraic structurePlanetmathPlanetmath, and in this sense the algebraic–analytic correspondence between the ramification points may be regarded as the geometric version of the equality (1) in number theory.

The algebraic–analytic correspondence of ramification points is itself only one manifestation of the wide ranging identification between algebraic geometry and analytic geometry which is explained to great effect in the seminal paper of Serre [6].


  • 1 Robin Hartshorne, Algebraic GeometryMathworldPlanetmathPlanetmath, Springer–Verlag, 1977 (GTM 52).
  • 2 Gerald Janusz, Algebraic Number Fields, Second Edition, American Mathematical Society, 1996 (GSM 7).
  • 3 Jürgen Jost, Compact Riemann Surfaces, Springer–Verlag, 1997.
  • 4 Dino Lorenzini, An Invitation to Arithmetic Geometry, American Mathematical Society, 1996 (GSM 9).
  • 5 Jean–Pierre Serre, Local FieldsMathworldPlanetmath, Springer–Verlag, 1979 (GTM 67).
  • 6 Jean–Pierre Serre, “Géométrie algébraique et géométrie analytique,” Ann. de L’Inst. Fourier 6 pp. 1–42, 1955–56.
  • 7 Joseph Silverman, The Arithmetic of Elliptic Curves, Springer–Verlag, 1986 (GTM 106).
Title ramification index
Canonical name RamificationIndex
Date of creation 2013-03-22 12:36:36
Last modified on 2013-03-22 12:36:36
Owner djao (24)
Last modified by djao (24)
Numerical id 17
Author djao (24)
Entry type Definition
Classification msc 11S15
Classification msc 30F99
Classification msc 30F99
Classification msc 12F99
Classification msc 13B02
Classification msc 14E22
Synonym ramify
Synonym ramified
Synonym unramified
Synonym ramification degree
Synonym ramification
Related topic NumberField
Related topic DecompositionGroup
Related topic UnramifiedExtensionsAndClassNumberDivisibility
Related topic SplittingAndRamificationInNumberFieldsAndGaloisExtensions
Defines branch point
Defines ramification point