inertial degree

inertial degree

Let

ι

:

A

→

B

be a

ring homomorphism

. Let

𝔓

⊂

B

be a

prime ideal

, with

𝔭

:=

ι

-

1

⁢

(

𝔓

)

⊂

A

. The

algebra

map

ι

induces an

A

/

𝔭

module

structure

on the ring

B

/

𝔓

. If the dimension of

B

/

𝔓

as an

A

/

𝔭

module exists, then it is called the

inertial degree

of

𝔓

over

A

.

A particular case of special importance in

number theory

is when

L

/

K

is a

field extension

and

ι

:

𝒪

K

→

𝒪

L

is the

inclusion map

of the

ring of integers

. In this case, the domain

𝒪

K

/

𝔭

is a field, so

dim

𝒪

K

/

𝔭

⁡

𝒪

L

/

𝔓

is guaranteed to exist, and the inertial degree of

𝔓

over

𝒪

K

is denoted

f

⁢

(

𝔓

/

𝔭

)

. We have the

formula

∑

𝔓

∣

𝔭

e

(

𝔓

/

𝔭

)

f

(

𝔓

/

𝔭

)

=

[

L

:

K

]

,

where

e

⁢

(

𝔓

/

𝔭

)

is the

ramification index

of

𝔓

over

𝔭

and the sum is taken over all prime ideals

𝔓

of

𝒪

L

dividing

𝔭

⁢

𝒪

L

. The prime

𝔓

(and also the prime

𝔭

) is said to be

inert

if

f

(

𝔓

/

𝔭

)

=

[

L

:

K

]

.

Example:

Let

ι

:

ℤ

→

ℤ

⁢

[

i

]

be the inclusion of the

integers

into the

Gaussian integers

. A prime

p

in

ℤ

may or may not factor in

ℤ

⁢

[

i

]

; if it does factor, then it must factor as

p

=

(

x

+

y

⁢

i

)

⁢

(

x

-

y

⁢

i

)

for some integers

x

,

y

. Thus a prime

p

factors into two primes if it equals

x

2

+

y

2

, and remains prime in

ℤ

⁢

[

i

]

otherwise. There are then three

categories

of primes in

ℤ

⁢

[

i

]

:

1.

The prime 2 factors as

(

1

+

i

)

⁢

(

1

-

i

)

, and the

principal ideals

generated by

(

1

+

i

)

and

(

1

-

i

)

are equal in

ℤ

⁢

[

i

]

, so the ramification index of

(

1

+

i

)

over

ℤ

is two. The ring

Z

⁢

[

i

]

/

(

1

+

i

)

is

isomorphic

to

ℤ

/

2

, so the inertial degree

f

⁢

(

(

1

+

i

)

/

(

2

)

)

is one.

2.

For primes

p

≡

1

mod

4

, the prime

p

∈

ℤ

factors into the

product

of the two primes

(

x

+

y

⁢

i

)

⁢

(

x

-

y

⁢

i

)

, with ramification index and inertial degree one.

3.

For primes

p

≡

3

(

mod

4

)

, the prime

p

remains prime in

ℤ

⁢

[

i

]

and

ℤ

⁢

[

i

]

/

(

p

)

is a two dimensional field extension of

ℤ

/

p

, so the inertial degree is two and the ramification index is one.

In all cases, the sum of the products of the inertial degree and ramification index is equal to 2, which is the dimension of the corresponding

extension

ℚ

⁢

(

i

)

/

ℚ

of

number fields

.

1

Local interpretations & generalizations

For any extension

ι

:

A

→

B

of

Dedekind domains

, the inertial degree of the prime

𝔓

⊂

B

over the prime

𝔭

:=

ι

-

1

⁢

(

𝔓

)

⊂

A

is equal to the inertial degree of

𝔓

⁢

B

𝔓

over

𝔭

⁢

A

𝔭

in the

localizations

at

𝔓

and

𝔭

. Moreover, the same is true even if we pass to

completions

of the

local rings

B

𝔓

and

A

𝔭

at

𝔓

and

𝔭

. The preservation of inertial degree and ramification indices with respect to localization is one of the reasons why the technique of localization is a useful tool in the study of such domains.

As in the case of ramification indices, it is possible to define the notion of inertial degree in the more general setting of

locally ringed spaces

. However, the

generalizations

of inertial degree are not as widely used because in

algebraic geometry

one usually works with a fixed

base field

, which makes all the

residue fields

at the points equal to the same field.

Title

inertial degree

Canonical name

InertialDegree

Date of creation

2013-03-22 12:38:17

Last modified on

2013-03-22 12:38:17

Owner

djao (24)

Last modified by

djao (24)

Numerical id

6

Author

djao (24)

Entry type

Definition

Classification

msc 12F99

Classification

msc 13B02

Classification

msc 11S15

Synonym

residue degree

Related topic

Ramify

Related topic

DecompositionGroup

Defines

inert

Generated on Fri Feb 9 16:44:31 2018 by

LaTeXML