infinite Galois theory

infinite Galois theory

Let

L

/

F

be a

Galois extension

, not necessarily finite dimensional.

1

Topology on the Galois group

Recall that the

Galois group

G

:=

Gal

⁡

(

L

/

F

)

of

L

/

F

is the
group of all

field automorphisms

σ

:

L

⟶

L

that restrict to
the

identity map

on

F

, under the

group operation

of

composition

. In
the case where the

extension

L

/

F

is infinite dimensional, the group

G

comes equipped with a natural topology, which plays a key role in
the statement of the

Galois correspondence

.

We define a subset

U

of

G

to be open if, for each

σ

∈

U

,
there exists an intermediate field

K

⊂

L

such that

•

The degree

[

K

:

F

]

is finite,

•

If

σ

′

is another element of

G

, and the

restrictions

σ

|

K

and

σ

′

|

K

are equal, then

σ

′

∈

U

.

The resulting

collection

of

open sets

forms a

topology

on

G

, called
the

Krull topology

, and

G

is a

topological group

under the
Krull topology. Another way to define the topology is to state that
the

subgroups

Gal

⁡

(

L

/

K

)

for

finite extensions

K

/

F

form a

neighborhood

basis for

Gal

⁡

(

L

/

F

)

at the

identity

.

2

Inverse limit structure

In this

section

we exhibit the group

G

as a

projective limit

of an

inverse system

of

finite groups

. This construction shows that the
Galois group

G

is actually a

profinite group

.

Let

𝒜

denote the set of finite

normal extensions

K

of

F

which
are

contained

in

L

. The set

𝒜

is a

partially ordered set

under
the inclusion relation. Form the

inverse limit

Γ

:=

lim

⟵

⁢

Gal

⁡

(

K

/

F

)

⊂

∏

K

∈

𝒜

Gal

⁡

(

K

/

F

)

consisting, as usual, of the set of all

(

σ

K

)

∈

∏

K

Gal

⁡

(

K

/

F

)

such that

σ

K

′

|

K

=

σ

K

for all

K

,

K

′

∈

𝒜

with

K

⊂

K

′

. We make

Γ

into a

topological space

by
putting the

discrete topology

on each

finite set

Gal

⁡

(

K

/

F

)

and
giving

Γ

the

subspace topology

induced

by the

product topology

on

∏

K

Gal

⁡

(

K

/

F

)

. The group

Γ

is a

closed subset

of the
compact group

∏

K

Gal

⁡

(

K

/

F

)

, and is therefore

compact

.

Let

ϕ

:

G

⟶

∏

K

∈

𝒜

Gal

⁡

(

K

/

F

)

be the

group homomorphism

which sends an

element

σ

∈

G

to the
element

(

σ

K

)

of

∏

K

Gal

⁡

(

K

/

F

)

whose

K

–th coordinate
is the

automorphism

σ

|

K

∈

Gal

⁡

(

K

/

F

)

. Then the function

ϕ

has image equal to

Γ

and in fact is a

homeomorphism

between

G

and

Γ

. Since

Γ

is profinite, it follows that

G

is profinite as well.

3

The Galois correspondence

Theorem 1

(Galois correspondence for infinite extensions).

Let

G

,

L

,

F

be as before. For every

closed subgroup

H

of

G

,
let

L

H

denote the

fixed field

of

H

. The correspondence

K

↦

Gal

⁡

(

L

/

K

)

,

defined for all intermediate field extensions

F

⊂

K

⊂

L

,
is an

inclusion

reversing

bijection

between the set of all
intermediate extensions

K

and the set of all closed subgroups of

G

. Its

inverse

is the correspondence

H

↦

L

H

,

defined for all closed subgroups

H

of

G

. The extension

K

/

F

is
normal if and only if

Gal

⁡

(

L

/

K

)

is a

normal subgroup

of

G

, and in
this case the restriction map

G

⟶

Gal

⁡

(

K

/

F

)

has kernel

Gal

⁡

(

L

/

K

)

.

Theorem 2

(Galois correspondence for finite subextensions).

Let

G

,

L

,

F

be as before.

•

Every

open subgroup

H

⊂

G

is closed and has finite index
in

G

.

•

If

H

⊂

G

is an open subgroup, then the field extension

L

H

/

F

is finite.

•

For every intermediate field

K

with

[

K

:

F

]

finite, the Galois
group

Gal

⁡

(

L

/

K

)

is an open subgroup of

G

.

Title

infinite Galois theory

Canonical name

InfiniteGaloisTheory

Date of creation

2013-03-22 12:39:06

Last modified on

2013-03-22 12:39:06

Owner

djao (24)

Last modified by

djao (24)

Numerical id

7

Author

djao (24)

Entry type

Topic

Classification

msc 12F10

Classification

msc 13B05

Related topic

FundamentalTheoremOfGaloisTheory

Related topic

GaloisGroup

Related topic

InverseLimit

Defines

Krull topology

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