ramification index

ramification index

1

Ramification in number fields

Definition 1

(First definition).

Let

L

/

K

be an

extension

of

number fields

. Let

𝔭

be a nonzero

prime ideal

in the

ring of integers

𝒪

K

of

K

, and suppose the
ideal

𝔭

⁢

𝒪

L

⊂

𝒪

L

factors as

𝔭

⁢

𝒪

L

=

∏

i

=

1

n

𝔓

i

e

i

for some prime ideals

𝔓

i

⊂

𝒪

L

and

exponents

e

i

∈

ℕ

. The

natural number

e

i

is called the

ramification index

of

𝔓

i

over

𝔭

. It is often denoted

e

⁢

(

𝔓

i

/

𝔭

)

. If

e

i

>

1

for any

i

, then we say the ideal

𝔭

ramifies

in

L

.

Likewise, if

𝔓

is a nonzero prime ideal in

𝒪

L

, and

𝔭

:=

𝔓

∩

𝒪

K

, then we say

𝔓

ramifies

over

K

if the
ramification index

e

⁢

(

𝔓

/

𝔭

)

of

𝔓

in the

factorization

of the
ideal

𝔭

⁢

𝒪

L

⊂

𝒪

L

is greater than 1. That is, a prime

𝔭

in

𝒪

K

ramifies in

L

if at least one prime

𝔓

dividing

𝔭

⁢

𝒪

L

ramifies over

K

. If

L

/

K

is a

Galois extension

, then the
ramification indices of all the primes dividing

𝔭

⁢

𝒪

L

are equal,
since the

Galois group

is

transitive

on this set of primes.

1.1

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

𝔭

:=

𝔓

∩

𝒪

K

. Then the induced map of

localizations

(

𝒪

K

)

𝔭

↪

(

𝒪

L

)

𝔓

is a local

homomorphism

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

𝔭

⁢

(

𝒪

L

)

𝔓

=

(

𝔓

⁢

(

𝒪

L

)

𝔓

)

e

⊂

(

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

ι

:

A

⟶

B

be any

ring homomorphism

. Suppose

𝔓

⊂

B

is a prime ideal such that the localization

B

𝔓

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

ι

𝔓

:

A

𝔭

⟶

B

𝔓

. Then the

ramification index

e

⁢

(

𝔓

/

𝔭

)

is
defined to be the unique natural number such that

ι

⁢

(

𝔭

)

⁢

B

𝔓

=

(

𝔓

⁢

B

𝔓

)

e

⁢

(

𝔓

/

𝔭

)

⊂

B

𝔓

,

or

∞

if

ι

⁢

(

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

/

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

algebraic

setting, but we limit ourselves
to the following remarks:

1.

Suppose

A

and

B

are themselves discrete valuation rings,
with respective

maximal ideals

𝔭

and

𝔓

. Let

A

^

:=

lim

⟵

⁢

A

/

𝔭

n

and

B

^

:=

lim

⟵

⁢

B

/

𝔓

n

be the

completions

of

A

and

B

with respect to

𝔭

and

𝔓

. Then

e

⁢

(

𝔓

/

𝔭

)

=

e

⁢

(

𝔓

⁢

B

^

/

𝔭

⁢

A

^

)

.

(1)

In other words, the ramification index of

𝔓

over

𝔭

in the

A

–

algebra

B

equals the ramification index in the completions of

A

and

B

with respect to

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

∑

𝔓

∣

𝔭

e

(

𝔓

/

𝔭

)

f

(

𝔓

/

𝔭

)

≤

[

L

:

K

]

,

(2)

where

𝔓

ranges over all prime ideals in

B

that divide

𝔭

⁢

B

, and

f

⁢

(

𝔓

/

𝔭

)

:=

dim

A

/

𝔭

⁡

(

B

/

𝔓

)

is the

inertial degree

of

𝔓

over

𝔭

.

Equality

holds in

Equation

(

2

) whenever

B

is

finitely generated

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

section

that the mathematical term
lives up to its common English meaning.

Definition 3

(Algebraic version).

Let

f

:

C

1

⟶

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:

•

The number of points in a

generic

fiber

f

-

1

⁢

(

p

)

, for

p

∈

C

2

•

The maximum number of points in

f

-

1

⁢

(

p

)

, for

p

∈

C

2

•

The degree of the extension

k

⁢

(

C

1

)

/

f

*

⁢

k

⁢

(

C

2

)

of

function
fields

There is a

finite set

of points

p

∈

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

∈

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

.

•

𝔭

=

𝔪

p

, the maximal ideal in

A

consisting of all

functions

which vanish at

p

.

•

𝔓

=

𝔪

P

, the maximal ideal in

B

consisting of all
functions which vanish at

P

.

•

ι

=

f

p

*

:

k

⁢

[

C

2

]

p

↪

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

=

ℂ

and

C

1

=

C

2

=

ℂ

=

𝔸

ℂ

1

. Take the map

f

:

ℂ

⟶

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

.

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.

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 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

]

⟶

k

⁢

[

C

1

]

of coordinate rings.

Equation (

2

) in this case says

∑

P

∈

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

:

ℂ

⟶

ℂ

be given by

f

⁢

(

y

)

=

y

2

as in
Example

4

. Since

C

2

is just the affine line, the
coordinate ring

ℂ

⁢

[

C

2

]

is equal to

ℂ

⁢

[

X

]

, the

polynomial ring

in
one

variable

over

ℂ

. Likewise,

ℂ

⁢

[

C

1

]

=

ℂ

⁢

[

Y

]

, and the induced
map

f

*

:

ℂ

⁢

[

X

]

⟶

ℂ

⁢

[

Y

]

is naturally given by

f

*

⁢

(

X

)

=

Y

2

. We
may accordingly identify the coordinate ring

ℂ

⁢

[

C

2

]

with the

subring

ℂ

⁢

[

X

2

]

of

ℂ

⁢

[

X

]

=

ℂ

⁢

[

C

1

]

.

Now, the ring

ℂ

⁢

[

X

]

is a

principal ideal domain

, and the maximal
ideals in

ℂ

⁢

[

X

]

are exactly the

principal ideals

of the form

(

X

-

a

)

for any

a

∈

ℂ

. Hence the nonzero prime ideals in

ℂ

⁢

[

X

2

]

are of the form

(

X

2

-

a

)

, and these factor in

ℂ

⁢

[

X

]

as

(

X

2

-

a

)

=

(

X

-

a

)

⁢

(

X

+

a

)

⊂

ℂ

⁢

[

X

]

.

Note that the two prime ideals

(

X

-

a

)

and

(

X

+

a

)

of

ℂ

⁢

[

X

]

are equal only when

a

=

0

, so we see that the ideal

(

X

2

-

a

)

in

ℂ

⁢

[

X

2

]

, corresponding to the point

a

∈

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

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

)

⟶

(

Y

,

𝒪

Y

)

be a

morphism of locally ringed
spaces

. Let

p

∈

X

and suppose that the stalk

(

𝒪

X

)

p

is a
discrete valuation ring. Write

ϕ

p

:

(

𝒪

Y

)

f

⁢

(

p

)

⟶

(

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

∞

if it does not exist), such that

ϕ

p

⁢

(

𝔪

f

⁢

(

p

)

)

⁢

(

𝒪

X

)

p

=

𝔪

p

e

,

where

𝔪

p

and

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

ι

:

A

⟶

B

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

:

C

1

⟶

C

2

, there is an induced
morphism

f

#

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

).

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

:

M

⟶

N

be a holomorphic map of

Riemann surfaces

. For any

p

∈

M

, there exists local coordinate charts

U

and

V

around

p

and

f

⁢

(

p

)

such that

f

is locally the map

z

↦

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

:

ℂ

⟶

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

=

2

⁢

w

+

w

2

. If we change coordinates to

z

=

2

⁢

w

+

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

↦

z

e

with

e

=

2

, so the ramification index of

f

at

y

=

0

is
equal to 2.

3.1

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

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

⟶

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

]

.

References

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

).

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

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