splitting and ramification in number fields and Galois extensions

splitting and ramification in number fields and Galois extensions

Let

F

/

K

be an

extension

of

number fields

and let

𝒪

F

and

𝒪

K

be their respective

rings of integers

. The ring of integers of a number field is a

Dedekind domain

, and these enjoy the property that every ideal

𝔄

factors uniquely as a finite product of

prime ideals

(see the entry

fractional ideal

(

http://planetmath.org/FractionalIdeal

)). Let

𝔭

be a prime ideal of

𝒪

K

. Then

𝔭

⁢

𝒪

F

is an ideal of

𝒪

F

. Let us assume that the prime ideal factorization of

𝔭

⁢

𝒪

F

into primes of

𝒪

F

is as follows:

𝔭

⁢

𝒪

F

=

∏

i

=

1

r

𝔓

i

e

i

(1)

We say that the primes

𝔓

i

lie above

𝔭

and

𝔓

i

|

𝔭

(divides). The

exponent

e

i

(commonly denoted as

e

(

𝔓

i

|

𝔭

)

) is the

ramification index

of

𝔓

i

over

𝔭

. Notice that for each prime ideal

𝔓

i

, the

quotient ring

𝒪

F

/

𝔓

i

is a

finite field extension

of the

finite field

𝒪

K

/

𝔭

(also called the

residue field

). The degree of this extension is called the

inertial degree

of

𝔓

i

over

𝔭

and it is usually denoted by:

f

(

𝔓

i

|

𝔭

)

=

[

𝒪

F

/

𝔓

i

:

𝒪

K

/

𝔭

]

.

Notice that as it is pointed out in the entry “inertial degree (

http://planetmath.org/InertialDegree

)”, the ramification index and the inertial degree are related by the formula:

∑

i

=

1

r

e

(

𝔓

i

|

𝔭

)

f

(

𝔓

i

|

𝔭

)

=

[

F

:

K

]

(2)

where

r

is the number of prime ideals lying above

𝔭

(as in Eq. (

1

)). See the theorem below for an improvement of Eq. (

2

) in the case when

F

/

K

is Galois.

Definition 1

.

Let

F

,

K

and

P

i

,

p

be as above.

1.

If

e

i

>

1

for some

i

, then we say that

𝔓

i

is

ramified

over

𝔭

and

𝔭

ramifies in

F

/

K

. If

e

i

=

1

for all

i

then we say that

𝔭

is unramified in

F

/

K

.

2.

If there is a

unique prime

ideal

𝔓

lying above

𝔭

(so

r

=

1

) and

f

(

𝔓

|

𝔭

)

=

1

then we say that

𝔭

is

totally ramified

in

F

/

K

. In this case

e

(

𝔓

|

𝔭

)

=

[

F

:

K

]

.

3.

On the other hand, if

e

(

𝔓

i

|

𝔭

)

=

f

(

𝔓

i

|

𝔭

)

=

1

for all

i

, we say that

𝔭

is

totally split

(or splits completely) in

F

/

K

. Notice that there are exactly

r

=

[

F

:

K

]

prime ideals of

𝒪

F

lying above

𝔭

.

4.

Let

p

be the

characteristic

of the residue field

𝒪

K

/

𝔭

. If

e

i

=

e

(

𝔓

i

|

𝔭

)

>

1

and

e

i

and

p

are relatively prime, then we say that

𝔓

i

is

tamely ramified

. If

p

|

e

i

then we say that

𝔓

i

is

strongly ramified

(or wildly ramified).

When the extension

F

/

K

is a

Galois extension

then Eq. (

2

) is quite more simple:

Theorem 1

.

Assume that

F

/

K

is a Galois extension of number fields. Then all the ramification indices

e

i

=

e

(

P

i

|

p

)

are equal to the same number

e

, all the inertial degrees

f

i

=

f

(

P

i

|

p

)

are equal to the same number

f

and the ideal

p

⁢

O

F

factors as:

𝔭

⁢

𝒪

F

=

∏

i

=

1

r

𝔓

i

e

=

(

𝔓

1

⋅

𝔓

2

⋅

…

⋅

𝔓

r

)

e

Moreover:

e

⋅

f

⋅

r

=

[

F

:

K

]

.

Title

splitting and ramification in number fields and Galois extensions

Canonical name

SplittingAndRamificationInNumberFieldsAndGaloisExtensions

Date of creation

2013-03-22 15:05:29

Last modified on

2013-03-22 15:05:29

Owner

alozano (2414)

Last modified by

alozano (2414)

Numerical id

11

Author

alozano (2414)

Entry type

Definition

Classification

msc 12F99

Classification

msc 13B02

Classification

msc 11S15

Synonym

completely split

Synonym

strongly ramified

Synonym

wild ramification

Related topic

Ramify

Related topic

InertialDegree

Related topic

CalculatingTheSplittingOfPrimes

Related topic

PrimeIdealDecompositionInQuadraticExtensionsOfMathbbQ

Related topic

PrimeIdealDecompositionInCyclotomicExtensionsOfMathbbQ

Defines

totally ramified

Defines

totally split

Defines

wildly ramified

Defines

tamely ramified

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