characterization of abelian extensions of exponent n

characterization of abelian extensions of exponent n

Theorem 1

.

Let

K

be a field containing the

n

th

roots of unity

, with

characteristic

not dividing

n

. Let

L

be a

finite extension

of

K

. Then the following are

equivalent

:

1.

L

/

K

is Galois with

abelian

Galois group

of

exponent

dividing

n

2.

L

=

K

⁢

(

a

1

n

,

…

,

a

k

n

)

for some

a

i

∈

K

.

Proof.

Let

ζ

∈

K

be a

primitive

n

th

root of unity.

(

2

⇒

1

)

:

Choose

α

i

∈

L

such that

α

i

n

=

a

i

∈

K

. Then for each

i

, the

elements

α

i

,

ζ

⁢

α

i

,

…

,

ζ

n

-

1

⁢

α

i

are distinct and are all the roots of

x

n

-

a

i

in

L

. Thus

x

n

-

a

i

is

separable

over

K

and splits in

L

, so that

L

is the

splitting field

of the set of

polynomials

{

x

n

-

a

i

∣

1

≤

i

≤

k

}

. Thus

L

/

K

is Galois. Given

σ

∈

Gal

⁡

(

L

/

K

)

, for each

i

we have

σ

⁢

(

α

i

)

=

ζ

j

⁢

α

i

for some

1

≤

j

≤

n

, so that

σ

k

⁢

(

α

i

)

=

ζ

k

⁢

j

⁢

α

i

. It follows that

σ

n

is the

identity

for every

σ

∈

Gal

⁡

(

L

/

K

)

, so that the exponent of

Gal

⁡

(

L

/

K

)

divides

n

. It remains to show that

Gal

⁡

(

L

/

K

)

is abelian; this follows trivially from the simple definition of the Galois action as

multiplication

by some

n

th

root of unity: if

σ

,

τ

∈

Gal

⁡

(

L

/

K

)

with

σ

⁢

(

α

i

)

=

ζ

r

⁢

α

i

,

τ

⁢

(

α

i

)

=

ζ

s

⁢

α

i

, then

(

σ

⁢

τ

)

⁢

(

α

i

)

=

σ

⁢

(

ζ

s

⁢

α

i

)

=

ζ

s

⁢

ζ

r

⁢

α

i

(

τ

⁢

σ

)

⁢

(

α

i

)

=

τ

⁢

(

ζ

r

⁢

α

i

)

=

ζ

r

⁢

ζ

s

⁢

α

i

Thus

σ

⁢

τ

=

τ

⁢

σ

on each

α

i

. But the

α

i

generate

L

/

K

, so

σ

⁢

τ

=

τ

⁢

σ

on

L

and

Gal

⁡

(

L

/

K

)

is abelian.

(

1

⇒

2

)

:

Let

G

=

Gal

⁡

(

L

/

K

)

, and write

G

=

C

1

×

…

×

C

r

where each

C

i

is cyclic;

|

C

i

|

=

m

i

∣

n

for each

i

. For each

i

, define a

subgroup

H

i

≤

G

by

H

i

=

C

1

×

…

×

C

i

-

1

×

C

i

+

1

×

…

×

C

r

Then

G

/

H

i

≅

C

i

. Let

L

i

be the

fixed field

of

H

i

.

L

i

is normal over

K

since

H

i

is normal in

G

, and

Gal

⁡

(

L

i

/

K

)

≅

G

/

H

i

≅

C

i

and thus

L

i

/

K

is cyclic Galois of order

m

i

.

K

contains the primitive

m

i

th

root of unity

ζ

n

/

m

i

and thus

L

i

=

K

⁢

(

α

i

)

for some

α

i

∈

L

with

α

i

m

i

∈

K

(by

Kummer theory

). But then also

α

i

n

∈

K

. Then

Gal

⁡

(

L

:

K

⁢

(

α

1

,

…

,

α

r

)

)

=

H

1

∩

…

∩

H

r

=

{

1

}

since any element of the left-hand group fixes each

α

i

and thus fixes

L

i

so is the identity in

G

/

H

i

. Thus

L

=

K

⁢

(

α

1

,

…

,

α

r

)

.
∎

Corollary 2

.

If

L

/

K

is the

maximal

abelian extension

of

K

of exponent

n

, where

n

is prime to the characteristic of

K

, then

L

=

K

⁢

(

{

a

n

}

)

for some set of

a

∈

K

.

Proof.

Clearly

K

(

{

a

n

∣

a

∈

K

*

)

is an

infinite

abelian extension of exponent

n

. If

L

is the maximal such

extension

, choose

b

∈

L

. Then

K

⁢

(

b

)

is a finite extension of exponent dividing

n

and thus

K

⁢

(

b

)

is of the required form. Thus

L

=

∪

b

∈

L

K

⁢

(

b

)

is also of the required form; for example, if

S

⊂

K

*

is a set of coset representatives for

K

*

/

(

K

*

)

n

, then

L

=

K

⁢

(

S

)

.
∎

References

1

Morandi, P.,

Field and

Galois Theory

, Springer, 1996.

Title

characterization of abelian extensions of exponent n

Canonical name

CharacterizationOfAbelianExtensionsOfExponentN

Date of creation

2013-03-22 18:42:13

Last modified on

2013-03-22 18:42:13

Owner

rm50 (10146)

Last modified by

rm50 (10146)

Numerical id

4

Author

rm50 (10146)

Entry type

Theorem

Classification

msc 12F10

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