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ramification index (Definition)

Ramification in number fields

Definition 1 (First definition)   Let $ L/K$ be an extension of number fields. Let $ {\mathfrak{p}}$ be a nonzero prime ideal in the ring of integers $ \O _K$ of $ K$, and suppose the ideal $ {\mathfrak{p}}\O _L \subset \O _L$ factors as
$\displaystyle {\mathfrak{p}}\O _L = \prod_{i=1}^n \P _i^{e_i} $
for some prime ideals $ \P _i \subset \O _L$ and exponents $ e_i \in \mathbb{N}$. The natural number $ e_i$ is called the ramification index of $ \P _i$ over $ {\mathfrak{p}}$. It is often denoted $ e(\P _i/{\mathfrak{p}})$. If $ e_i > 1$ for any $ i$, then we say the ideal $ {\mathfrak{p}}$ ramifies in $ L$.

Likewise, if $ \P$ is a nonzero prime ideal in $ \O _L$, and $ {\mathfrak{p}}:= \P\cap \O _K$, then we say $ \P$ ramifies over $ K$ if the ramification index $ e(\P /{\mathfrak{p}})$ of $ \P$ in the factorization of the ideal $ {\mathfrak{p}}\O _L \subset \O _L$ is greater than 1. That is, a prime $ {\mathfrak{p}}$ in $ \O _K$ ramifies in $ L$ if at least one prime $ \P$ dividing $ {\mathfrak{p}} \O _L$ ramifies over $ K$. If $ L/K$ is a Galois extension, then the ramification indices of all the primes dividing $ {\mathfrak{p}}\O _L$ are equal, since the Galois group is transitive on this set of primes.

The local view

The phenomenon of ramification has an equivalent interpretation in terms of local rings. With $ L/K$ as before, let $ \P$ be a prime in $ \O _L$ with $ {\mathfrak{p}}:= \P\cap \O _K$. Then the induced map of localizations $ (\O _K)_{\mathfrak{p}}\hookrightarrow (\O _L)_\P$ is a local homomorphism of local rings (in fact, of discrete valuation rings), and the ramification index of $ \P$ over $ {\mathfrak{p}}$ is the unique natural number $ e$ such that

$\displaystyle {\mathfrak{p}}(\O _L)_\P = (\P (\O _L)_\P )^e \subset (\O _L)_\P . $

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 $ \iota: A \longrightarrow B$ be any ring homomorphism. Suppose $ \P\subset B$ is a prime ideal such that the localization $ B_\P$ of $ B$ at $ \P$ is a discrete valuation ring. Let $ {\mathfrak{p}}$ be the prime ideal $ \iota^{-1}(\P ) \subset A$, so that $ \iota$ induces a local homomorphism $ \iota_\P : A_{\mathfrak{p}}\longrightarrow B_\P$. Then the ramification index $ e(\P /{\mathfrak{p}})$ is defined to be the unique natural number such that
$\displaystyle \iota({\mathfrak{p}})B_\P = (\P B_\P )^{e(\P /{\mathfrak{p}})} \subset B_\P , $
or $ \infty$ if $ \iota({\mathfrak{p}})B_\P = (0)$.

The reader who is not interested in local rings may assume that $ A$ and $ B$ are unique factorization domains, in which case $ e(\P /{\mathfrak{p}})$ is the exponent of $ \P$ in the factorization of the ideal $ \iota({\mathfrak{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 algebraic setting, but we limit ourselves to the following remarks:

  1. Suppose $ A$ and $ B$ are themselves discrete valuation rings, with respective maximal ideals $ {\mathfrak{p}}$ and $ \P$. Let $ \hat{A} := \,\underset{\longleftarrow}{\lim}\, A/{\mathfrak{p}}^n$ and $ \hat{B} := \,\underset{\longleftarrow}{\lim}\,B/\P ^n$ be the completions of $ A$ and $ B$ with respect to $ {\mathfrak{p}}$ and $ \P$. Then
    $\displaystyle e(\P /{\mathfrak{p}}) = e(\P\hat{B}/{\mathfrak{p}}\hat{A}).$ (1)

    In other words, the ramification index of $ \P$ over $ {\mathfrak{p}}$ in the $ A$-algebra $ B$ equals the ramification index in the completions of $ A$ and $ B$ with respect to $ {\mathfrak{p}}$ and $ \P$.
  2. Suppose $ A$ and $ B$ are Dedekind domains, with respective fraction fields $ K$ and $ L$. If $ B$ equals the integral closure of $ A$ in $ L$, then
    $\displaystyle \sum_{\P\div {\mathfrak{p}}} e(\P /{\mathfrak{p}}) f(\P /{\mathfrak{p}}) \leq [L:K],$ (2)

    where $ \P$ ranges over all prime ideals in $ B$ that divide $ {\mathfrak{p}}B$, and $ f(\P /{\mathfrak{p}}) := \dim_{A/{\mathfrak{p}}}(B/\P )$ is the inertial degree of $ \P$ over $ {\mathfrak{p}}$. Equality holds in Equation (2) whenever $ B$ is finitely generated as an $ A$-module.

Ramification in algebraic geometry

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

Definition 3 (Algebraic version)   Let $ f: C_1 \longrightarrow C_2$ be a non-constant regular morphism of curves (by which we mean one dimensional nonsingular irreducible algebraic varieties) over an algebraically closed field $ k$. Then $ f$ has a nonzero degree $ n := \deg f$, which can be defined in any of the following ways:

There is a finite set of points $ p \in C_2$ for which the inverse image $ f^{-1}(p)$ does not have size $ n$, and we call these points the branch points or ramification points of $ f$. If $ P \in C_1$ 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[C_2]_p$, the local ring consisting of all rational functions in the function field $ k(C_2)$ which are regular at $ p$.
  • $ B = k[C_1]_P$, the local ring consisting of all rational functions in the function field $ k(C_1)$ which are regular at $ P$.
  • $ {\mathfrak{p}}= {\mathfrak{m}}_p$, the maximal ideal in $ A$ consisting of all functions which vanish at $ p$.
  • $ \P = {\mathfrak{m}}_P$, the maximal ideal in $ B$ consisting of all functions which vanish at $ P$.
  • $ \iota = f^*_p: k[C_2]_p \hookrightarrow k[C_1]_P$, the map on the function fields induced by the morphism $ f$.
Example 4   The picture in Figure 1 may be worth a thousand words. Let $ k = \mathbb{C}$ and $ C_1 = C_2 = \mathbb{C}= \mathbb{A}^1_\mathbb{C}$. Take the map $ f: \mathbb{C}\longrightarrow \mathbb{C}$ given by $ f(y) = y^2$. Then $ f$ is plainly a map of degree 2, and every point in $ C_2$ except for 0 has two preimages in $ C_1$. 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) = y^2$ near $ y=0$.
\begin{figure}\epsfig{file=curve.eps}\end{figure}

Note that we have only drawn the real locus of $ f$ because that is all that can fit into two dimensions. We see from the figure that a typical point on $ C_2$ such as the point $ x = 1$ has two points in $ C_1$ which map to it, but that the point $ x = 0$ has only one corresponding point of $ C_1$ which “branches” or “ramifies” into two distinct points of $ C_1$ whenever one moves away from 0.

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 varieties. When $ C_1$ and $ C_2$ are affine, then their coordinate rings $ k[C_1]$ and $ k[C_2]$ are Dedekind domains, and the points of the curve $ C_1$ (respectively, $ C_2$) correspond naturally with the maximal ideals of the ring $ k[C_1]$ (respectively, $ k[C_2]$). The ramification points of the curve $ C_1$ are then exactly the points of $ C_1$ which correspond to maximal ideals of $ k[C_1]$ that ramify in the algebraic sense, with respect to the map $ f^*: k[C_2] \longrightarrow k[C_1]$ of coordinate rings.

Equation (2) in this case says

$\displaystyle \sum_{P \in f^{-1}(p)} e(P/p) = n, $
and we see that the well known formula (2) in number theory is simply the algebraic analogue of the geometric fact that the number of points in the fiber of $ f$, counting multiplicities, is always $ n$.
Example 5   Let $ f: \mathbb{C}\longrightarrow \mathbb{C}$ be given by $ f(y) = y^2$ as in Example 4. Since $ C_2$ is just the affine line, the coordinate ring $ \mathbb{C}[C_2]$ is equal to $ \mathbb{C}[X]$, the polynomial ring in one variable over $ \mathbb{C}$. Likewise, $ \mathbb{C}[C_1] = \mathbb{C}[Y]$, and the induced map $ f^*: \mathbb{C}[X] \longrightarrow \mathbb{C}[Y]$ is naturally given by $ f^*(X) = Y^2$. We may accordingly identify the coordinate ring $ \mathbb{C}[C_2]$ with the subring $ \mathbb{C}[X^2]$ of $ \mathbb{C}[X] = \mathbb{C}[C_1]$.

Now, the ring $ \mathbb{C}[X]$ is a principal ideal domain, and the maximal ideals in $ \mathbb{C}[X]$ are exactly the principal ideals of the form $ (X - a)$ for any $ a \in \mathbb{C}$. Hence the nonzero prime ideals in $ \mathbb{C}[X^2]$ are of the form $ (X^2 - a)$, and these factor in $ \mathbb{C}[X]$ as

$\displaystyle (X^2 - a) = (X - \sqrt{a}) (X + \sqrt{a}) \subset \mathbb{C}[X]. $
Note that the two prime ideals $ (X - \sqrt{a})$ and $ (X + \sqrt{a})$ of $ \mathbb{C}[X]$ are equal only when $ a = 0$, so we see that the ideal $ (X^2 - a)$ in $ \mathbb{C}[X^2]$, corresponding to the point $ a \in C_2$, ramifies in $ C_1$ 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 $ \mathbb{C}[X]$.

In the case where $ f$ is a map between projective varieties, Definition 2 does not directly apply to the coordinate rings of $ C_1$ and $ C_2$, but only to those of open covers of $ C_1$ and $ C_2$ 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,\O _X) \longrightarrow (Y,\O _Y)$ be a morphism of locally ringed spaces. Let $ p \in X$ and suppose that the stalk $ (\O _X)_p$ is a discrete valuation ring. Write $ \phi_p: (\O _Y)_{f(p)} \longrightarrow (\O _X)_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 $ \infty$ if it does not exist), such that
$\displaystyle \phi_p({\mathfrak{m}}_{f(p)})(\O _X)_p = {\mathfrak{m}}_p^e, $
where $ {\mathfrak{m}}_p$ and $ {\mathfrak{m}}_{f(p)}$ are the respective maximal ideals of $ (\O _X)_p$ and $ (\O _Y)_{f(p)}$. We say $ p$ is ramified in $ Y$ if $ e > 1$.
Example 7   A ring homomorphism $ \iota: A \longrightarrow B$ corresponds functorially to a morphism $ \operatorname{Spec}(B) \longrightarrow \operatorname{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: C_1 \longrightarrow C_2$, there is an induced morphism $ f^\char93 $ on the structure sheaves of $ C_1$ and $ C_2$, which are locally ringed spaces. If $ C_1$ and $ C_2$ are curves, then the stalks are one dimensional regular local rings and therefore discrete valuation rings, so in this way we recover the algebraic geometric definition (Definition 3) from the sheaf definition (Definition 6).

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: M \longrightarrow N$ be a holomorphic map of Riemann surfaces. For any $ p \in M$, there exists local coordinate charts $ U$ and $ V$ around $ p$ and $ f(p)$ such that $ f$ is locally the map $ z \mapsto z^e$ 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: \mathbb{C}\longrightarrow \mathbb{C}$, $ f(y) = y^2$ 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 + w^2$. If we change coordinates to $ z = 2w + w^2$ 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) = y^2$ is already in the form $ z \mapsto z^e$ with $ e=2$, so the ramification index of $ f$ at $ y=0$ is equal to 2.

Algebraic-analytic correspondence

Of course, the analytic 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 functions is a locally ringed space. Furthermore the stalk at any point is always a discrete valuation ring, because germs of holomorphic functions have Taylor expansions making the stalk isomorphic to the power series ring $ \mathbb{C}[[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: V \longrightarrow W$ 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 structure, 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].

Bibliography

1
Robin Hartshorne, Algebraic Geometry, 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 Fields, 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).



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See Also: number field, decomposition group, unramified extensions and class number divisibility, splitting and ramification in number fields and Galois extensions

Other names:  ramify, ramified, unramified, ramification degree, ramification
Also defines:  branch point, ramification point

Attachments:
splitting and ramification in number fields and Galois extensions (Definition) by alozano
ramification of archimedean places (Definition) by alozano
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Cross-references: analytic geometry, algebraic geometry, algebraic structure, analytic structure, power series, isomorphic, Taylor expansions, germs, analytic, domain, behavior, charts, local coordinate, Riemann surfaces, holomorphic, geometry, complex, regular local rings, sheaves, structure, prime spectrum, locally ringed spaces, stalk, morphism of locally ringed spaces, straight, jump, open covers, projective varieties, characterization, principal ideals, principal ideal domain, subring, variable, polynomial ring, line, multiplicities, number theory, coordinate, affine varieties, dimensions, locus, real, near, graph, index, preimages, thousand, morphism, vanish, functions, regular, rational functions, size, inverse image, finite set, function fields, fiber, generic, points, number, degree, field, algebraically closed, varieties, irreducible, nonsingular, mean, curves, regular morphism, section, finitely generated, equation, equality, inertial degree, ranges, integral closure, fraction fields, Dedekind domains, completions, maximal ideals, limit, algebraic, rings, unique factorization domains, induces, ring homomorphism, even, discrete valuation rings, homomorphism, localizations, map, induced, local rings, terms, interpretation, equivalent, transitive, Galois group, indices, Galois extension, prime, natural number, exponents, factors, ideal, ring of integers, prime ideal, number fields, extension
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This is version 14 of ramification index, born on 2002-04-23, modified 2005-03-15.
Object id is 2868, canonical name is Ramify.
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AMS MSC: 11S15 (Number theory :: Algebraic number theory: local and $p$-adic fields :: Ramification and extension theory)
 14E22 (Algebraic geometry :: Birational geometry :: Ramification problems)
 13B02 (Commutative rings and algebras :: Ring extensions and related topics :: Extension theory)
 12F99 (Field theory and polynomials :: Field extensions :: Miscellaneous)
 30F99 (Functions of a complex variable :: Riemann surfaces :: Miscellaneous)
 30F99 (Functions of a complex variable :: Riemann surfaces :: Miscellaneous)
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