aarjaneiro/planetmath huggingface dataset:
perfect field

perfect field

A

perfect field

is a field

K

such that every

algebraic extension

field

L

/

K

is

separable

over

K

.

All fields of

characteristic

0 are perfect, so in particular the fields

ℝ

,

ℂ

and

ℚ

are perfect. If

K

is a field of characteristic

p

(with

p

a

prime number

), then

K

is perfect if and only if the

Frobenius endomorphism

F

on

K

, defined by

F

(

x

)

=

x

p

(

x

∈

K

)

,

is an

automorphism

of

K

. Since the

Frobenius map

is always

injective

, it is

sufficient

to check whether

F

is

surjective

. In particular, all

finite fields

are perfect (any injective endomorphism is also surjective). Moreover, any field whose characteristic is nonzero that is

algebraic

(

http://planetmath.org/AlgebraicExtension

) over its

prime subfield

is perfect. Thus, the only fields that are not perfect are those whose characteristic is nonzero and are transcendental over their prime subfield.

Similarly, a ring

R

of characteristic

p

is perfect if the endomorphism

x

↦

x

p

of

R

is an

automorphism

(i.e., is surjective).

Title

perfect field

Canonical name

PerfectField

Date of creation

2013-03-22 13:08:23

Last modified on

2013-03-22 13:08:23

Owner

sleske (997)

Last modified by

sleske (997)

Numerical id

11

Author

sleske (997)

Entry type

Definition

Classification

msc 12F10

Related topic

SeparablePolynomial

Related topic

ExtensionField

Defines

perfect

Defines

perfect ring

Generated on Fri Feb 9 16:43:48 2018 by

LaTeXML
proof of additive form of Hilbert’s theorem 90

proof of additive form of Hilbert’s theorem 90

Set

n

=

[

L

:

K

]

.

First, let there be

x

∈

L

such that

y

=

x

-

σ

⁢

(

x

)

. Then

Tr

⁡

(

y

)

=

(

x

-

σ

⁢

(

x

)

)

+

(

σ

⁢

(

x

)

-

σ

2

⁢

(

x

)

)

+

⋯

+

(

σ

n

-

1

⁢

(

x

)

-

σ

n

⁢

(

x

)

)

=

0

because

x

=

σ

n

⁢

(

x

)

.

Now, let

Tr

⁡

(

y

)

=

0

. Choose

z

∈

L

with

Tr

⁡

(

z

)

≠

0

. Then there exists

x

∈

L

with

x

⁢

Tr

⁡

(

z

)

=

y

⁢

σ

⁢

(

z

)

+

(

y

+

σ

⁢

(

y

)

)

⁢

σ

2

⁢

(

z

)

+

⋯

+

(

y

+

σ

⁢

(

y

)

+

⋯

+

σ

n

-

1

⁢

(

y

)

)

⁢

σ

n

-

1

⁢

(

z

)

.

Since

Tr

⁡

(

z

)

∈

K

we have

σ

⁢

(

x

)

⁢

Tr

⁡

(

z

)

=

σ

⁢

(

y

)

⁢

σ

2

⁢

(

z

)

+

(

σ

⁢

(

y

)

+

σ

2

⁢

(

y

)

)

⁢

σ

3

⁢

(

z

)

+

⋯

+

(

σ

⁢

(

y

)

+

⋯

+

σ

n

-

2

)

⁢

σ

n

-

1

⁢

(

z

)

+

(

σ

⁢

(

y

)

+

⋯

+

σ

n

-

1

⁢

(

y

)

)

⁢

σ

n

⁢

(

z

)

.

Now remember that

Tr

⁡

(

y

)

=

0

. We obtain

(

x

-

σ

⁢

(

x

)

)

⁢

Tr

⁡

(

z

)

=

y

⁢

σ

⁢

(

z

)

+

(

y

+

σ

⁢

(

y

)

)

⁢

σ

2

⁢

(

z

)

+

⋯

+

(

y

+

σ

⁢

(

y

)

+

⋯

+

σ

n

-

1

⁢

(

y

)

)

⁢

σ

n

-

1

⁢

(

z

)

-

σ

⁢

(

y

)

⁢

σ

2

⁢

(

z

)

-

(

σ

⁢

(

y

)

+

σ

2

⁢

(

y

)

)

⁢

σ

3

⁢

(

z

)

-

⋯

-

(

σ

⁢

(

y

)

+

⋯

+

σ

n

-

2

)

⁢

σ

n

-

1

⁢

(

z

)

+

y

⁢

z

=

y

⁢

Tr

⁡

(

z

)

,

so

y

=

x

-

σ

⁢

(

x

)

, as we wanted to show.

Title

proof of additive form of Hilbert’s theorem 90

Canonical name

ProofOfAdditiveFormOfHilbertsTheorem90

Date of creation

2013-03-22 15:21:25

Last modified on

2013-03-22 15:21:25

Owner

mathwizard (128)

Last modified by

mathwizard (128)

Numerical id

4

Author

mathwizard (128)

Entry type

Proof

Classification

msc 12F10

Classification

msc 11R32

Generated on Fri Feb 9 16:43:50 2018 by

LaTeXML
proof of characterization of perfect fields

proof of characterization of perfect fields

Proposition 1

The following are

equivalent

:

(a)

Every

algebraic extension

of

K

is

separable

.

(b)

Either

char

⁡

K

=

0

or

char

⁡

K

=

p

and the

Frobenius map

is

surjective

.

Proof.

Suppose (a) and not (b). Then we must have

char

⁡

K

=

p

>

0

, and there must be

a

∈

K

with no

p

-th root in

K

. Let

L

be a

splitting field

over

K

for the

polynomial

X

p

-

a

,
and let

α

∈

L

be a root of this polynomial.
Then

(

X

-

α

)

p

=

X

p

-

α

p

=

X

p

-

a

, which has coefficients in

K

.
This means that the minimum polynomial for

α

over

K

must be a divisor of

(

X

-

α

)

p

and so must have repeated roots. This is not possible since

L

is separable
over

K

.

Conversely

, suppose (b) and not (a).
Let

α

be an

element

which is algebraic over

K

but not separable.
Then its minimum polynomial

f

must have a repeated root, and by replacing

α

by this
root if necessary, we may assume that

α

is a repeated root of

f

.
Now,

f

′

has coefficients in

K

and also has

α

as a root. Since it is of lower
degree than

f

, this is not possible unless

f

′

=

0

,
whence

char

⁡

K

=

p

>

0

and

f

has the form:

f

=

x

p

⁢

n

+

a

n

-

1

⁢

x

p

⁢

(

n

-

1

)

+

…

+

a

1

⁢

x

p

+

a

0

.

with

a

o

≠

0

.
By (b), we may choose elements

b

i

∈

K

, for

0

≤

i

≤

n

-

1

such that

b

i

p

=

a

i

.
Then we may write

f

as:

f

=

(

x

n

+

b

n

-

1

⁢

x

n

-

1

+

…

+

b

1

⁢

x

+

b

0

)

p

.

Since

f

⁢

(

α

)

=

0

and since the Frobenius map

x

↦

x

p

is

injective

, we see that

α

n

+

b

n

-

1

⁢

α

n

-

1

+

…

+

b

1

⁢

α

+

b

0

=

0

But then

α

is a root of the polynomial

x

n

+

b

n

-

1

⁢

x

n

-

1

+

…

+

b

1

⁢

x

+

b

0

which has coefficients in

K

, is non-zero (since

b

o

≠

0

), and has lower degree than

f

.
This contradicts the choice of

f

as the minimum polynomial of

α

.

Title

proof of characterization of perfect fields

Canonical name

ProofOfCharacterizationOfPerfectFields

Date of creation

2013-03-22 14:47:44

Last modified on

2013-03-22 14:47:44

Owner

mclase (549)

Last modified by

mclase (549)

Numerical id

6

Author

mclase (549)

Entry type

Proof

Classification

msc 12F10

Generated on Fri Feb 9 16:43:55 2018 by

LaTeXML
radical extension

radical extension

A

radical tower

is a

field extension

L

/

F

which has a

filtration

F

=

L

0

⊂

L

1

⊂

⋯

⊂

L

n

=

L

where for each

i

,

0

≤

i

<

n

, there exists an

element

α

i

∈

L

i

+

1

and a

natural number

n

i

such that

L

i

+

1

=

L

i

⁢

(

α

i

)

and

α

i

n

i

∈

L

i

.

A

radical extension

is a field extension

K

/

F

for which there exists a radical tower

L

/

F

with

L

⊃

K

. The notion of radical extension coincides with the informal

concept

of solving for the roots of a

polynomial

by

radicals

, in the sense that a polynomial over

K

is

solvable by radicals

if and only if its

splitting field

is a radical extension of

F

.

Title

radical extension

Canonical name

RadicalExtension

Date of creation

2013-03-22 12:08:35

Last modified on

2013-03-22 12:08:35

Owner

djao (24)

Last modified by

djao (24)

Numerical id

7

Author

djao (24)

Entry type

Definition

Classification

msc 12F10

Synonym

radical tower

Generated on Fri Feb 9 16:44:04 2018 by

LaTeXML
separable

separable

An

irreducible polynomial

f

∈

F

⁢

[

x

]

with

coefficients

in a field

F

is

separable

if

f

factors into distinct linear factors over a

splitting field

K

of

f

.

A

polynomial

g

with coefficients in

F

is

separable

if each

irreducible

factor of

g

in

F

⁢

[

x

]

is a

separable polynomial

.

An

algebraic field extension

K

/

F

is

separable

if, for each

a

∈

K

, the

minimal polynomial

of

a

over

F

is separable. When

F

has characteristic zero, every

algebraic extension

of

F

is separable; examples of inseparable

extensions

include the quotient field

K

⁢

(

u

)

⁢

[

t

]

/

(

t

p

-

u

)

over the field

K

⁢

(

u

)

of

rational functions

in one

variable

, where

K

has

characteristic

p

>

0

.

More generally, an arbitrary field extension

K

/

F

is defined to be

separable

if every

finitely generated

intermediate field extension

L

/

F

has a transcendence basis

S

⊂

L

such that

L

is a separable algebraic extension of

F

⁢

(

S

)

.

Title

separable

Canonical name

Separable

Date of creation

2013-03-22 12:08:04

Last modified on

2013-03-22 12:08:04

Owner

djao (24)

Last modified by

djao (24)

Numerical id

13

Author

djao (24)

Entry type

Definition

Classification

msc 12F10

Classification

msc 11R32

Related topic

PerfectField

Defines

separable

Defines

separable polynomial

Defines

separable extension

Generated on Fri Feb 9 16:44:07 2018 by

LaTeXML
separable closure

separable closure

Let

K

be a field and let

L

be an

algebraic closure

of

K

. The

separable closure

of

K

inside

L

is the

compositum

of all finite

separable extensions

of

K

contained in

L

(that is to say, the smallest

subfield

of

L

that contains every finite separable extension of

K

).

Title

separable closure

Canonical name

SeparableClosure

Date of creation

2013-03-22 12:12:11

Last modified on

2013-03-22 12:12:11

Owner

djao (24)

Last modified by

djao (24)

Numerical id

6

Author

djao (24)

Entry type

Definition

Classification

msc 12F10

Generated on Fri Feb 9 16:44:09 2018 by

LaTeXML
solvable by radicals

solvable by radicals

Let

F

be a field and

f

⁢

(

x

)

∈

F

⁢

[

x

]

. Then

f

⁢

(

x

)

is

solvable by radicals

if there exists a

radical extension

K

/

F

such that

f

⁢

(

x

)

factors into linears when considered as an element of

K

⁢

[

x

]

.

Title

solvable by radicals

Canonical

name

SolvableByRadicals

Date of creation

2013-03-22 16:55:39

Last modified on

2013-03-22 16:55:39

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

5

Author

Wkbj79 (1863)

Entry type

Definition

Classification

msc 12F10

Synonym

solvability by radicals

Synonym

solvable polynomial

Related topic

Radical5

Related topic

RadicalExtension

Generated on Fri Feb 9 16:44:11 2018 by

LaTeXML
purely inseparable

purely inseparable

Let

F

be a field of

characteristic

p

>

0

and let

α

be an

element

which is algebraic over

F

. Then

α

is said to be

purely inseparable

over

F

if

α

p

n

∈

F

for some

n

≥

0

.

An

algebraic field extension

K

/

F

is

purely inseparable

if each element of

K

is purely inseparable over

F

.

Purely inseparable extensions

have the following property: if

K

/

F

is purely inseparable, and

A

is an

algebraic closure

of

F

which contains

K

, then any

homomorphism

K

→

A

which fixes

F

necessarily fixes

K

.

Let

K

/

F

be an arbitrary

algebraic extension

. Then there is an intermediate field

E

such that

K

/

E

is purely inseparable, and

E

/

F

is

separable

.

Example

.

Let

s

be an indeterminate, and let

K

=

𝔽

3

⁢

(

s

)

where

𝔽

3

is the

finite field

with

3

elements. Let

F

=

𝔽

3

⁢

(

s

6

)

. Then

K

/

F

is neither separable, nor purely inseparable. Let

E

=

𝔽

3

⁢

(

s

3

)

. Then

E

/

F

is separable, and

K

/

E

is purely inseparable.

Title

purely inseparable

Canonical name

PurelyInseparable

Date of creation

2013-03-22 14:49:08

Last modified on

2013-03-22 14:49:08

Owner

mclase (549)

Last modified by

mclase (549)

Numerical id

6

Author

mclase (549)

Entry type

Definition

Classification

msc 12F15

Generated on Fri Feb 9 16:44:13 2018 by

LaTeXML
topics on field extensions and Galois theory

topics on field extensions and Galois theory

Definitions

1.

extension field

2.

simple field extension

3.

degree (

http://

planetmath

.org/Degree

)

4.

field adjunction

5.

composite field

6.

finite extension

7.

algebraic extension

8.

algebraic closure

9.

abelian number field

10.

abelian extension

11.

biquadratic field

12.

biquadratic extension

13.

quadratic closure

14.

cyclic extension

15.

pure cubic field

16.

number field

17.

algebraically solvable

18.

norm and trace of algebraic number

19.

perfect field

20.

cyclotomic field

21.

cyclotomic extension

22.

p

-

extension

(

http://planetmath.org/PExtension

)

23.

radical

24.

radical extension

25.

solvable by radicals

26.

expressible

27.

normal extension

28.

normal closure

29.

separable extension

30.

separable closure

31.

splitting field

32.

fixed field

of a set of

automorphisms

33.

conjugate fields

34.

Galois extension

35.

Galois closure

36.

Galois group

37.

absolute Galois group

38.

Galois connection

39.

inverse Galois problem

40.

function field

41.

ray class field

Field theory in relation to Euclidean geometry

1.

Euclidean field

2.

constructible numbers

3.

motivation of definition of constructible numbers

4.

compass and straightedge construction

5.

theorem on constructible numbers

6.

theorem on constructible angles

7.

criterion for constructibility of regular polygon

8.

classical problems of constructibility

Galois theory

1.

fundamental theorem of Galois theory

2.

Galois group of the compositum of two Galois extensions

3.

Galois

subfields

of real radical extensions (

http://planetmath.org/GaloisSubfieldsOfRealRootExtensionsAreAtMostQuadratic

)

4.

compositum of a Galois extension and another extension (

http://planetmath.org/CompositumOfAGaloisExtensionAndAnotherExtensionIsGalois

)

5.

criterion for solvability of a

polynomial

by radicals (

http://planetmath.org/GaloisCriterionForSolvabilityOfAPolynomialByRadicals

)

6.

Hilbert Theorem 90

7.

Shafarevich’s theorem (finite

solvable groups

as Galois groups)

8.

number fields with a particular finite

abelian

Galois group (

http://planetmath.org/GaloisGroupsOfFiniteAbelianExtensionsOfMathbbQ

)

Applications of Galois theory

1.

Galois group of a biquadratic extension

2.

cubic formula

3.

quartic formula

4.

Galois group of a

cubic polynomial

5.

Galois group of a

quartic polynomial

6.

Galois-theoretic

derivation

of the cubic formula

7.

proof of fundamental theorem of algebra

8.

casus irreducibilis

9.

Cardano’s formulae

10.

a

condition of algebraic extension

11.

primitive element theorem

(Steinitz)

Title

topics on field extensions and Galois theory

Canonical name

TopicsOnFieldExtensionsAndGaloisTheory

Date of creation

2013-03-22 17:43:11

Last modified on

2013-03-22 17:43:11

Owner

rm50 (10146)

Last modified by

rm50 (10146)

Numerical id

13

Author

rm50 (10146)

Entry type

Topic

Classification

msc 12F15

Classification

msc 12F10

Classification

msc 12F05

Generated on Fri Feb 9 16:44:15 2018 by

LaTeXML
linearly disjoint

linearly disjoint

Let

E

and

F

be

subfields

of

L

, each containing a field

K

.

E

is said to be

linearly disjoint

from

F

over

K

if every subset of

E

linearly independent

over

K

is also linearly independent over

F

.

Remark.

If

E

is linearly disjoint from

F

over

K

, then

F

is linearly disjoint from

E

over

K

. Then one can speak of

E

and

F

being linearly disjoint over

K

without causing any confusions.

Title

linearly disjoint

Canonical name

LinearlyDisjoint

Date of creation

2013-03-22 14:19:28

Last modified on

2013-03-22 14:19:28

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

7

Author

CWoo (3771)

Entry type

Definition

Classification

msc 12F20

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

LaTeXML
transcendence degree

transcendence degree

The

transcendence degree

of a set

S

over a field

K

, denoted

T

S

, is the size of the maximal subset

S

′

of

S

such that all the elements of

S

′

are

algebraically independent

.

The

transcendence degree

of a

field extension

L

over

K

is the transcendence degree of the

minimal

subset of

L

needed to generate

L

over

K

.

Heuristically speaking, the transcendence degree of a

finite set

S

is obtained by taking the number of

elements

in the set, subtracting the number of algebraic elements in that set, and then subtracting the number of

algebraic

relations

between distinct pairs of elements in

S

.

Example 1

(Computing the Transcendence Degree).

The set

S

=

{

7

,

π

,

π

2

,

e

}

has transcendence

T

S

≤

2

over

Q

since there are
four elements,

7

is algebraic, and the

polynomial

f

⁢

(

x

,

y

)

=

x

2

-

y

gives an algebraic dependence between

π

and

π

2

(i.e.

(

π

,

π

2

)

is a root of

f

), giving

T

S

≤

4

-

1

-

1

=

2

. If
we assume the

conjecture

that

e

and

π

are algebraically
independent, then no more dependencies can exist, and we can conclude
that, in fact,

T

S

=

2

.

Title

transcendence degree

Canonical

name

TranscendenceDegree

Date of creation

2013-03-22 13:58:11

Last modified on

2013-03-22 13:58:11

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

7

Author

mathcam (2727)

Entry type

Definition

Classification

msc 12F20

Defines

transcendence degree of a set

Defines

transcendence degree of a field extension

Generated on Fri Feb 9 16:44:18 2018 by

LaTeXML
composite field

composite field

Let

{

K

α

}

,

α

∈

J

, be a

collection

of

subfields

of a field

L

. The

composite field

of the collection is the smallest subfield of

L

that contains all the fields

K

α

.

The notation

K

1

⁢

K

2

(resp.,

K

1

⁢

K

2

⁢

…

⁢

K

n

) is often used to denote the composite field of two (resp., finitely many) fields.

Title

composite field

Canonical name

CompositeField

Date of creation

2013-03-22 12:12:09

Last modified on

2013-03-22 12:12:09

Owner

djao (24)

Last modified by

djao (24)

Numerical id

6

Author

djao (24)

Entry type

Definition

Classification

msc 12F99

Synonym

compositum

Synonym

composite extension

Generated on Fri Feb 9 16:44:19 2018 by

LaTeXML
corollary to the compositum of a Galois extension and another extension is Galois

corollary to the compositum of a Galois extension and another extension is Galois

Corollary 1

.

Let

E

/

K

be a

Galois extension

of fields, let

F

/

K

be an arbitrary

extension

and assume that

E

and

F

are both

subfields

of some other larger field

T

. The

compositum

of

E

and

F

is here denoted by

E

⁢

F

. Then

[

E

F

:

F

]

=

[

E

:

E

∩

F

]

.

This follows immediately from item (2) of the theorem.

Title

corollary to the compositum of a Galois extension and another extension is Galois

Canonical name

CorollaryToTheCompositumOfAGaloisExtensionAndAnotherExtensionIsGalois

Date of creation

2013-03-22 18:42:04

Last modified on

2013-03-22 18:42:04

Owner

rm50 (10146)

Last modified by

rm50 (10146)

Numerical id

4

Author

rm50 (10146)

Entry type

Corollary

Classification

msc 12F99

Classification

msc 11R32

Generated on Fri Feb 9 16:44:21 2018 by

LaTeXML
extension field

extension field

We say that a field

K

is an

extension

of

F

if

F

is a

subfield

of

K

.

We usually denote

K

being an extension of

F

by

F

⊂

K

,

F

≤

K

,

K

/

F

or

Generated on Fri Feb 9 16:44:25 2018 by

LaTeXML
field adjunction

field adjunction

Let

K

be a field and

E

an

extension field

of

K

.  If

α

∈

E

, then the smallest

subfield

of

E

, that contains

K

and

α

, is denoted by

K

⁢

(

α

)

.  We say that

K

⁢

(

α

)

is
obtained from the field

K

by

adjoining

the element

α

to

K

via

field adjunction

.

Theorem

.

K

⁢

(

α

)

is identical with the

quotient field

Q

of

K

⁢

[

α

]

.

Proof.

(1) Because

K

⁢

[

α

]

is an

integral domain

(as a

subring

of the field

E

), all possible

quotients

of the elements of

K

⁢

[

α

]

belong to

E

. So we have

K

∪

{

α

}

⊆

K

⁢

[

α

]

⊆

Q

⊆

E

,

and because

K

⁢

(

α

)

was the smallest, then

K

⁢

(

α

)

⊆

Q

.

(2)

K

⁢

(

α

)

is a subring of

E

containing

K

and

α

, therefore also the whole ring

K

⁢

[

α

]

, that is,

K

⁢

[

α

]

⊆

K

⁢

(

α

)

.  And because

K

⁢

(

α

)

is a field, it must contain all possible quotients of the elements of

K

⁢

[

α

]

, i.e.,

Q

⊆

K

⁢

(

α

)

.

In to the

adjunction

of one single element, we can adjoin to

K

an arbitrary subset

S

of

E

:  the resulting field

K

⁢

(

S

)

is the smallest of such subfields of

E

, i.e. the intersection of such subfields of

E

, that contain both

K

and

S

.  We say that

K

⁢

(

S

)

is obtained from

K

by adjoining the set

S

to it.  Naturally,

K

⊆

K

⁢

(

S

)

⊆

E

.

The field

K

⁢

(

S

)

contains all elements of

K

and

S

, and being a field, also all such elements that can be formed via

addition

,

subtraction

,

multiplication

and division from the elements of

K

and

S

.  But such elements constitute a field, which therefore must be equal with

K

⁢

(

S

)

.  Accordingly, we have the

Theorem.

K

⁢

(

S

)

constitutes of all rational expressions formed of the elements of the field

K

with the elements of the set

S

.

Notes.

1.

K

⁢

(

S

)

is the union of all fields

K

⁢

(

T

)

where

T

is a finite subset of

S

.

2.

K

⁢

(

S

1

∪

S

2

)

=

K

⁢

(

S

1

)

⁢

(

S

2

)

.

3. If, especially,

S

also is a subfield of

E

, then one may denote

K

⁢

(

S

)

=

K

⁢

S

.

References

1

B. L. van der Waerden:

Algebra

. Erster Teil

.  Siebte Auflage der

Modernen Algebra

. Springer-Verlag; Berlin, Heidelberg, New York (1966).

Title

field adjunction

Canonical

name

FieldAdjunction

Date of creation

2015-02-21 15:39:45

Last modified on

2015-02-21 15:39:45

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

16

Author

pahio (2872)

Entry type

Definition

Classification

msc 12F99

Synonym

simple extension

Related topic

GroundFieldsAndRings

Related topic

Forcing

Related topic

PolynomialRingOverFieldIsEuclideanDomain

Related topic

AConditionOfAlgebraicExtension

Generated on Fri Feb 9 16:44:27 2018 by

LaTeXML
Galois group of the compositum of two Galois extensions

Galois group of the compositum of two Galois extensions

Theorem 1

.

Let

E

and

F

be

Galois extensions

of a field

K

. Then:

1.

The

intersection

E

∩

F

is Galois over

K

.

2.

The

compositum

E

⁢

F

is Galois over

K

. Moreover, the

Galois group

Gal

⁡

(

E

⁢

F

/

K

)

is

isomorphic

to the

subgroup

H

of the

direct product

G

=

Gal

⁡

(

E

/

K

)

×

Gal

⁡

(

F

/

K

)

given by:

H

=

{

(

σ

,

ψ

)

:

σ

|

E

∩

F

=

ψ

|

E

∩

F

}

i. e.

H

consists of pairs of

elements of

G

whose

restrictions

to

E

∩

F

are equal.

Corollary 1

.

Let

E

and

F

be Galois extensions of a field

K

such that

E

∩

F

=

K

. Then

E

⁢

F

is Galois over

K

and the Galois group is isomorphic to the direct product:

Gal

⁡

(

E

⁢

F

/

K

)

≅

Gal

⁡

(

E

/

K

)

×

Gal

⁡

(

F

/

K

)

.

Title

Galois group of the compositum of two Galois extensions

Canonical name

GaloisGroupOfTheCompositumOfTwoGaloisExtensions

Date of creation

2013-03-22 15:04:22

Last modified on

2013-03-22 15:04:22

Owner

alozano (2414)

Last modified by

alozano (2414)

Numerical id

5

Author

alozano (2414)

Entry type

Theorem

Classification

msc 12F99

Classification

msc 11R32

Related topic

CompositumOfAGaloisExtensionAndAnotherExtensionIsGalois

Related topic

GaloisExtension

Generated on Fri Feb 9 16:44:29 2018 by

LaTeXML