ramification of archimedean places

ramification of archimedean places

Throughout this entry, if

α

is a

complex number

, we denote the

complex conjugate

of

α

by

α

¯

.

Definition 1

.

Let

K

be a

number field

.

1.

An

archimedean place

of

K

is either a

real embedding

ϕ

:

K

→

ℝ

or a pair of complex-conjugate

embeddings

(

ψ

,

ψ

¯

)

, with

ψ

¯

≠

ψ

and

ψ

:

K

→

ℂ

. The archimedean places are sometimes called the infinite places (cf.

place of field

).

2.

The

non-archimedean places

of

K

are the

prime ideals

in

𝒪

K

, the

ring of integers

of

K

(see

non-archimedean valuation

(

http://planetmath.org/

Valuation

)). The non-archimedean places are sometimes called the finite places.

Notice that any archimedean place

ϕ

:

K

→

ℂ

can be extended to an embedding

ϕ

^

:

ℚ

¯

→

ℂ

, where

ℚ

¯

is a fixed

algebraic closure

of

ℚ

(in order to prove this, one uses the fact that

ℂ

is

algebraically closed

and also Zorn’s Lemma). See also this entry (

http://planetmath.org/PlaceAsExtensionOfHomomorphism

). In particular, if

F

is a

finite extension

of

K

then

ϕ

can be extended to an archimidean place

ϕ

^

:

F

→

ℂ

of

F

.

Next, we define the decomposition and

inertia group

associated to archimedean places. For the case of non-archimedean places (i.e. prime ideals) see the entries

decomposition group

and

ramification

.

Let

F

/

K

be a finite

Galois extension

of number fields and let

ϕ

be a (real or a pair of

complex

) archimedean place of

K

. Let

ϕ

1

and

ϕ

2

be two archimedean places of

F

which extend

ϕ

. Notice that, since

F

/

K

is Galois, the image of

ϕ

1

and

ϕ

2

are equal, in other words:

ϕ

1

⁢

(

F

)

=

ϕ

2

⁢

(

F

)

⊂

ℂ

.

Hence, the

composition

ϕ

1

-

1

∘

ϕ

2

is an

automorphism

of

F

(here

ϕ

1

-

1

denotes the inverse map of

ϕ

1

, restricted to

ϕ

1

⁢

(

F

)

). Thus,

ϕ

1

-

1

∘

ϕ

2

=

σ

∈

Gal

⁡

(

F

/

K

)

and

ϕ

2

=

ϕ

1

∘

σ

so

ϕ

1

and

ϕ

2

differ by an element of the

Galois group

. Similarly, if

(

ψ

1

,

ψ

1

¯

)

and

(

ψ

2

,

ψ

2

¯

)

are complex embeddings which extend

ϕ

, then there is

σ

∈

Gal

⁡

(

F

/

K

)

such that

(

ψ

2

,

ψ

2

¯

)

=

(

ψ

1

,

ψ

1

¯

)

∘

σ

meaning that either

ψ

2

=

ψ

1

∘

σ

(and thus

ψ

2

¯

=

ψ

1

¯

∘

σ

) or

ψ

2

¯

=

ψ

1

∘

σ

(and thus

ψ

2

=

ψ

1

¯

∘

σ

). We are ready now to make the

definitions

.

Definition 2

.

Let

F

/

K

be a Galois extension of number fields and let

w

be an archimedean place of

F

lying above a place

v

of

K

. The decomposition and inertia

subgroups

for the pair

w

|

v

are equal and are defined by:

D

(

w

|

v

)

=

T

(

w

|

v

)

=

{

σ

∈

Gal

(

F

/

K

)

:

w

∘

σ

=

w

}

.

Let

e

=

e

(

w

|

v

)

=

|

T

(

w

|

v

)

|

be the size of the inertia subgroup. If

e

>

1

then we say that the

archimedean place

v

is ramified

in the

extension

F

/

K

.

The ramification in the

archimedean

case is much simpler than the non-archimedean analogue. One readily proves the following

proposition

:

Proposition 1

.

The inertia subgroup

T

(

w

|

v

)

is nontrivial only when

v

is real,

w

=

(

ψ

,

ψ

¯

)

is a complex archimedean place of

F

and

σ

is the “complex conjugation” map which has order

2

. Therefore

e

(

w

|

v

)

=

1

or

2

and ramification of archimedean places occurs if and only if there is a complex place of

F

lying above a real place of

K

.

Proof.

Suppose first that

w

=

ϕ

:

F

→

ℝ

is a real embedding. Then

ϕ

is injective and

ϕ

∘

σ

=

ϕ

implies that

σ

is the

identity

automorphism and

T

(

w

|

v

)

would be trivial. So let us assume that

w

=

(

ψ

,

ψ

¯

)

is a complex archimedean place and let

σ

∈

Gal

⁡

(

F

/

K

)

such that

(

ψ

,

ψ

¯

)

=

(

ψ

,

ψ

¯

)

∘

σ

.

Therefore, either

ψ

=

ψ

∘

σ

(which implies that

σ

is the identity by the injectivity of

ψ

) or

ψ

=

ψ

¯

∘

σ

. The latter implies that

σ

=

ψ

-

1

¯

∘

ψ

, which is simply complex conjugation:

ψ

-

1

¯

∘

ψ

⁢

(

k

)

=

ψ

-

1

⁢

(

ψ

⁢

(

k

)

)

¯

=

k

¯

.

Finally, since

w

is an extension of

v

, the

equation

w

∘

σ

=

w

restricts to

v

¯

=

v

, thus

v

must be real.
∎

Corollary 1

.

Suppose

L

/

K

is an extension of number fields and assume that

K

is a totally imaginary (

http://planetmath.org/TotallyRealAndImaginaryFields

) number field. Then the extension

L

/

K

is unramified at all archimedean places.

Proof.

Since

K

is totally imaginary none of the embeddings of

K

are real. By the proposition, only real places can ramify.
∎

Title

ramification of archimedean places

Canonical

name

RamificationOfArchimedeanPlaces

Date of creation

2013-03-22 15:07:19

Last modified on

2013-03-22 15:07:19

Owner

alozano (2414)

Last modified by

alozano (2414)

Numerical id

9

Author

alozano (2414)

Entry type

Definition

Classification

msc 12F99

Classification

msc 13B02

Classification

msc 11S15

Synonym

finite place

Synonym

infinite place

Related topic

DecompositionGroup

Related topic

PlaceOfField

Related topic

RealAndComplexEmbeddings

Related topic

PlaceAsExtensionOfHomomorphism

Defines

decomposition and inertia group for archimedean places

Defines

archimedean place

Defines

non-archimedean place

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