aarjaneiro/planetmath huggingface dataset:
trace

trace

Let

K

/

F

be a

Galois extension

, and let

x

∈

K

. The

trace

Tr

F

K

⁡

(

x

)

of

x

is defined to be the sum of all the elements of the orbit of

x

under the

group action

of the

Galois group

Gal

⁡

(

K

/

F

)

on

K

; taken with multiplicities if

K

/

F

is a

finite extension

.

In the case where

K

/

F

is a finite extension,

Tr

F

K

⁡

(

x

)

:=

∑

σ

∈

Gal

⁡

(

K

/

F

)

σ

⁢

(

x

)

The trace of

x

is always an element of

F

, since any element of

Gal

⁡

(

K

/

F

)

permutes the orbit of

x

and thus fixes

Tr

F

K

⁡

(

x

)

.

The name “trace” derives from the fact that, when

K

/

F

is finite, the trace of

x

is simply the trace of the

linear transformation

T

:

K

⟶

K

of

vector spaces

over

F

defined by

T

⁢

(

v

)

:=

x

⁢

v

.

Title

trace

Canonical

name

Trace1

Date of creation

2013-03-22 12:17:59

Last modified on

2013-03-22 12:17:59

Owner

djao (24)

Last modified by

djao (24)

Numerical id

7

Author

djao (24)

Entry type

Definition

Classification

msc 12F05

Generated on Fri Feb 9 16:42:58 2018 by

LaTeXML
Zariski lemma

Zariski lemma

Proposition 1

.

Let

R

⊆

S

⊆

T

be

commutative rings

. If

R

is

noetherian

, and T

finitely generated

as an

R

-

algebra

and as an

S

-module, then

S

is finitely generated as an

R

-algebra.

Lemma 1

(Zariski’s lemma).

Let

(

L

:

K

)

be a

field extension

and

a

1

,

…

,

a

n

∈

L

be such that

K

⁢

(

a

1

,

…

,

a

n

)

=

K

⁢

[

a

1

,

…

,

a

n

]

. Then the

elements

a

1

,

…

,

a

n

are algebraic over

K

.

Proof.

The case

n

=

1

is clear. Now suppose

n

>

1

and not all

a

i

,

1

≤

i

≤

n

are algebraic over

K

.

Wlog we may assume

a

1

,

…

,

a

n

are

algebraically independent

and each element

a

r

+

1

,

…

,

a

n

is algebraic over

D

:=

K

⁢

(

a

1

,

…

,

a

r

)

. Hence

K

⁢

[

a

1

,

…

,

a

n

]

is a finite

algebraic extension

of

D

and therefore is a finitely generated

D

-module.

The above

proposition

applied to

K

⊆

D

⊆

K

⁢

[

a

1

,

…

,

a

n

]

shows that

D

is finitely generated as a

K

-algebra, i.e

D

=

K

⁢

[

d

1

,

…

,

d

n

]

.

Let

d

i

=

p

i

⁢

(

a

1

,

…

,

a

n

)

q

i

⁢

(

a

1

,

…

,

a

n

)

, where

p

i

,

q

i

∈

K

⁢

[

x

1

,

…

,

x

n

]

.

Now

a

1

,

…

,

a

n

are algebraically independent so that

K

⁢

[

a

1

,

…

,

a

n

]

≅

K

⁢

[

x

1

,

…

,

x

n

]

, which is a UFD (

http://planetmath.org/UFD

).

Let

h

be a

prime divisor

of

q

1

⁢

⋯

⁢

q

r

+

1

. Since

q

is relatively prime to each of

q

i

, the element

q

⁢

(

a

1

,

…

,

a

n

)

-

1

∈

D

cannot be in

K

⁢

[

d

1

,

…

,

d

n

]

. We obtain a

contradiction

.
∎

Title

Zariski lemma

Canonical name

ZariskiLemma

Date of creation

2013-03-22 17:18:11

Last modified on

2013-03-22 17:18:11

Owner

polarbear (3475)

Last modified by

polarbear (3475)

Numerical id

7

Author

polarbear (3475)

Entry type

Derivation

Classification

msc 12F05

Classification

msc 11J85

Generated on Fri Feb 9 16:43:01 2018 by

LaTeXML
abelian extension

abelian extension

Let

K

be a

Galois extension

of

F

. The

extension

is said to be an

abelian extension

if the

Galois group

Gal

(

K

/

F

)

is

abelian

.

Examples:

ℚ

⁢

(

2

)

/

ℚ

has Galois group

ℤ

/

2

⁢

ℤ

so

ℚ

⁢

(

2

)

/

ℚ

is an abelian extension.

Let

ζ

n

be a primitive nth root of unity (

http://planetmath.org/RootOfUnity

). Then

ℚ

⁢

(

ζ

n

)

/

ℚ

has
Galois group

(

ℤ

/

n

⁢

ℤ

)

*

(the

group of units

of

ℤ

/

n

⁢

ℤ

) so

ℚ

⁢

(

ζ

n

)

/

ℚ

is abelian.

Title

abelian extension

Canonical name

AbelianExtension

Date of creation

2013-03-22 13:09:28

Last modified on

2013-03-22 13:09:28

Owner

scanez (1021)

Last modified by

scanez (1021)

Numerical id

5

Author

scanez (1021)

Entry type

Definition

Classification

msc 12F10

Related topic

KroneckerWeberTheorem

Related topic

KummerTheory

Generated on Fri Feb 9 16:43:03 2018 by

LaTeXML
absolute Galois group

absolute Galois group

Let

k

be a field. The

absolute Galois group

G

k

of

k

is the

Galois group

Gal

⁡

(

k

sep

/

k

)

of the

field extension

k

sep

/

k

, where

k

sep

is the

separable closure

of

k

.

Title

absolute Galois group

Canonical

name

AbsoluteGaloisGroup

Date of creation

2013-03-22 12:12:14

Last modified on

2013-03-22 12:12:14

Owner

djao (24)

Last modified by

djao (24)

Numerical id

7

Author

djao (24)

Entry type

Definition

Classification

msc 12F10

Generated on Fri Feb 9 16:43:04 2018 by

LaTeXML
algebraically solvable

algebraically solvable

An equation

x

n

+

a

1

⁢

x

n

-

1

+

…

+

a

n

=

0

,

(1)

with

coefficients

a

j

in a field

K

, is

algebraically
solvable

, if some of its roots (

http://planetmath.org/Equation

) may
be expressed with the

elements of

K

by using

rational

operations (

addition

,

subtraction

,

multiplication

,

division

)
and

root extractions

.  I.e., a root of (1) is in a field

K

⁢

(

ξ

1

,

ξ

2

,

…

,

ξ

m

)

which is obtained of

K

by adjoining (

http://planetmath.org/FieldAdjunction

) to it in
succession certain suitable

radicals

ξ

1

,

ξ

2

,

…

,

ξ

m

.  Each radical may
under the root sign one or more of
the previous radicals,

{

ξ

1

=

r

1

p

1

,

ξ

2

=

r

2

⁢

(

ξ

1

)

p

2

,

ξ

3

=

r

3

⁢

(

ξ

1

,

ξ

2

)

p

3

,

⋯

⋯

ξ

m

=

r

m

⁢

(

ξ

1

,

ξ

2

,

…

,

ξ

m

-

1

)

p

m

,

where generally

r

k

⁢

(

ξ

1

,

ξ

2

,

…

,

ξ

k

-

1

)

is an element of the field

K

⁢

(

ξ

1

,

ξ

2

,

…

,

ξ

k

-

1

)

but no

p

k

’th power of an element of this field.  Because of the formula

r

j

⁢

k

=

r

k

j

one can, without hurting the generality, suppose that the indices (

http://planetmath.org/Root

)

p

1

,

p

2

,

…

,

p

m

are

prime numbers

.

Example.

Cardano’s formulae show that all roots of the

cubic equation

y

3

+

p

⁢

y

+

q

=

0

are in the

algebraic number field

which is obtained by adjoining to the field

ℚ

⁢

(

p

,

q

)

successively the radicals

ξ

1

=

(

q

2

)

2

+

(

p

3

)

3

,

ξ

2

=

-

q

2

+

ξ

1

3

,

ξ

3

=

-

3

.

In fact, as we consider also the equation (4), the roots may be expressed as

{

y

1

=

ξ

2

-

p

3

⁢

ξ

2

y

2

=

-

1

+

ξ

3

2

⋅

ξ

2

-

-

1

-

ξ

3

2

⋅

p

3

⁢

ξ

2

y

3

=

-

1

-

ξ

3

2

⋅

ξ

2

-

-

1

+

ξ

3

2

⋅

p

3

⁢

ξ

2

References

1

K. Väisälä:

Lukuteorian ja korkeamman algebran alkeet

.  Tiedekirjasto No. 17.   Kustannusosakeyhtiö Otava, Helsinki (1950).

Title

algebraically solvable

Canonical

name

AlgebraicallySolvable

Date of creation

2015-04-15 13:48:08

Last modified on

2015-04-15 13:48:08

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

9

Author

pahio (2872)

Entry type

Definition

Classification

msc 12F10

Synonym

algebraic solvability

Synonym

solvable algebraically

Related topic

RadicalExtension

Related topic

KalleVaisala

Generated on Fri Feb 9 16:43:07 2018 by

LaTeXML
casus irreducibilis

casus irreducibilis

Let the

polynomial

P

⁢

(

x

)

:=

x

n

+

a

1

⁢

x

n

-

1

+

…

+

a

n

with

complex

coefficients

a

j

be

irreducible

(

http://planetmath.org/IrreduciblePolynomial2

), i.e. irreducible in the field

ℚ

⁢

(

a

1

,

…

,

a

n

)

of its coefficients.  If the equation

P

⁢

(

x

)

=

0

can be solved algebraically (

http://planetmath.org/AlgebraicallySolvable

) and if all of its roots are real, then no root may be expressed with the numbers

a

j

using mere real

radicals

(

http://planetmath.org/NthRoot

) unless the degree (

http://planetmath.org/AlgebraicEquation

)

n

of the equation is an

integer

power (

http://planetmath.org/GeneralAssociativity

) of 2.

References

1

K. Väisälä

:

Lukuteorian ja korkeamman algebran alkeet

.  Tiedekirjasto No. 17. Kustannusosakeyhtiö Otava, Helsinki (1950).

Title

casus irreducibilis

Canonical name

CasusIrreducibilis

Date of creation

2013-03-22 15:21:00

Last modified on

2013-03-22 15:21:00

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

12

Author

pahio (2872)

Entry type

Theorem

Classification

msc 12F10

Related topic

RadicalExtension

Related topic

CardanosFormulae

Related topic

TakingSquareRootAlgebraically

Related topic

EulersDerivationOfTheQuarticFormula

Generated on Fri Feb 9 16:43:09 2018 by

LaTeXML
cyclic extension

cyclic extension

A

Galois extension

K

/

F

is said to be a

cyclic extension

if the

Galois group

Gal

(

K

/

F

)

is cyclic.

Title

cyclic extension

Canonical name

CyclicExtension

Date of creation

2013-03-22 13:09:31

Last modified on

2013-03-22 13:09:31

Owner

scanez (1021)

Last modified by

scanez (1021)

Numerical id

5

Author

scanez (1021)

Entry type

Definition

Classification

msc 12F10

Related topic

KummerTheory

Generated on Fri Feb 9 16:43:13 2018 by

LaTeXML
equivalent conditions for normality of a field extension

equivalent conditions for normality of a field extension

Theorem

.

If

K

/

F

is an

algebraic extension

of fields, then the following are

equivalent

:

1.

K

is normal over

F

;

2.

K

is the

splitting field

over

F

of a set of

polynomials

in

F

⁢

[

X

]

;

3.

if

F

¯

is an

algebraic closure

F

containing

K

and

σ

:

K

→

F

¯

is an

F

-

monomorphism

, then

σ

⁢

(

K

)

=

K

.

Proof.

(1)

⇒

(2) Let

X

be an

F

-basis for

K

, and for each

x

∈

X

, let

f

x

be the

irreducible polynomial

of

x

over

F

. By

hypothesis

, each

f

x

splits over

K

, and because we evidently have

K

=

F

⁢

(

X

)

, it follows that

K

is a splitting field of

{

f

x

:

x

∈

X

}

over

F

.

(2)

⇒

(3) Assume that

K

is a splitting field over

F

of

S

⊆

F

⁢

[

X

]

. Given

f

∈

S

, we may write

f

⁢

(

X

)

=

u

⁢

∏

i

=

1

n

(

X

-

u

i

)

for some

u

,

u

1

,

…

,

u

n

∈

K

; because

σ

fixes

F

pointwise

, we have

σ

⁢

(

u

i

)

∈

{

u

1

,

…

,

u

n

}

for

1

≤

i

≤

n

, and since

σ

is

injective

, it must simply permute the roots of

f

. Thus

u

1

,

…

,

u

n

∈

σ

⁢

(

K

)

. As

K

is generated over

F

by the roots of the polynomials in

S

, we obtain

K

=

σ

⁢

(

K

)

.

(3)

⇒

(1) Let

K

¯

be an algebraic closure of

K

, noting that, since

K

is algebraic over

F

, that same is true of

K

¯

, and consequently

K

¯

is an algebraic closure of

F

containing

K

. Now suppose

f

∈

F

⁢

[

X

]

is

irreducible

and that

u

∈

K

is a root of

f

, and let

v

be any root of

F

in

K

¯

. There exists an

F

-

isomorphism

τ

:

F

⁢

(

u

)

→

F

⁢

(

v

)

such that

τ

⁢

(

u

)

=

v

. Because

K

¯

is a splitting field over both

F

⁢

(

u

)

and

F

⁢

(

v

)

of the set of irreducible polynomials in

F

⁢

[

X

]

,

τ

extends to an

F

-isomorphism

σ

:

K

¯

→

K

¯

. It follows that

σ

|

K

:

K

→

K

¯

is an

F

-monomorphism, so that, by hypothesis,

σ

⁢

(

K

)

=

K

, hence that

v

=

σ

⁢

(

u

)

∈

K

. Thus

f

splits over

K

, and therefore

K

/

F

is normal.
∎

Title

equivalent conditions for

normality

of a

field extension

Canonical name

EquivalentConditionsForNormalityOfAFieldExtension

Date of creation

2013-03-22 18:37:56

Last modified on

2013-03-22 18:37:56

Owner

azdbacks4234 (14155)

Last modified by

azdbacks4234 (14155)

Numerical id

8

Author

azdbacks4234 (14155)

Entry type

Theorem

Classification

msc 12F10

Related topic

NormalExtension

Related topic

SplittingField

Related topic

ExtensionField

Related topic

AlgebraicExtension

Generated on Fri Feb 9 16:43:15 2018 by

LaTeXML
example of an extension that is not normal

example of an extension that is not normal

In this entry,

2

3

indicates the real

cube root

of

2

.

Consider the

extension

ℚ

⁢

(

2

3

)

/

ℚ

. The

minimal polynomial

for

2

3

over

ℚ

is

x

3

-

2

. This

polynomial

factors in

ℚ

⁢

(

2

3

)

as

x

3

-

2

=

(

x

-

2

3

)

⁢

(

x

2

+

x

⁢

2

3

+

4

3

)

. Let

f

⁢

(

x

)

=

x

2

+

x

⁢

2

3

+

4

3

. Note that

disc

⁡

(

f

⁢

(

x

)

)

=

(

2

3

)

2

-

4

⁢

4

3

=

4

3

-

4

⁢

4

3

=

-

3

⁢

4

3

<

0

. Thus,

f

⁢

(

x

)

has no

real roots

. Therefore,

f

⁢

(

x

)

has no roots in

ℚ

⁢

(

2

3

)

since

ℚ

⁢

(

2

3

)

⊆

ℝ

. Hence,

x

3

-

2

has a root in

ℚ

⁢

(

2

3

)

but does not split in

ℚ

⁢

(

2

3

)

. It follows that the extension

ℚ

⁢

(

2

3

)

/

ℚ

is not normal.

Title

example of an extension that is not normal

Canonical name

ExampleOfAnExtensionThatIsNotNormal

Date of creation

2013-03-22 16:00:28

Last modified on

2013-03-22 16:00:28

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

6

Author

Wkbj79 (1863)

Entry type

Example

Classification

msc 12F10

Generated on Fri Feb 9 16:43:17 2018 by

LaTeXML
example of nonperfect field

example of nonperfect field

In this entry, we exhibit an example of a field that is not a

perfect field

.

Let

F

=

𝔽

p

⁢

(

t

)

, where

𝔽

p

is the field with

p

elements

and

t

transcendental

over

𝔽

p

. The

splitting field

E

of the

irreducible polynomial

f

=

x

p

-

t

is not

separable

over

F

. Indeed, if

α

is an element of

E

such that

α

p

=

t

, we have

x

p

-

t

=

x

p

-

α

p

=

(

x

-

α

)

p

,

which shows that

f

has one root of

multiplicity

p

.

Title

example of nonperfect field

Canonical name

ExampleOfNonperfectField

Date of creation

2013-03-22 13:08:31

Last modified on

2013-03-22 13:08:31

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

7

Author

CWoo (3771)

Entry type

Example

Classification

msc 12F10

Generated on Fri Feb 9 16:43:19 2018 by

LaTeXML
example of normal extension

example of normal extension

Let

F

=

ℚ

⁢

(

2

)

. Then the

extension

F

/

ℚ

is normal because

F

is clearly the

splitting field

of the

polynomial

f

⁢

(

x

)

=

x

2

-

2

. Furthermore

F

/

ℚ

is a

Galois extension

with

Gal

⁡

(

F

/

ℚ

)

≅

ℤ

/

2

⁢

ℤ

.

Now, let

2

1

/

4

denote the positive real fourth root of

2

and define

K

=

F

⁢

(

2

1

/

4

)

. Then the extension

K

/

F

is normal because

K

is the splitting field of

k

⁢

(

x

)

=

x

2

-

2

, and as before

K

/

F

is a Galois extension with

Gal

⁡

(

K

/

F

)

≅

ℤ

/

2

⁢

ℤ

.

However, the extension

K

/

ℚ

is neither normal nor Galois. Indeed, the polynomial

g

⁢

(

x

)

=

x

4

-

2

has one root in

K

(actually two), namely

2

1

/

4

, and yet

g

⁢

(

x

)

does not split in

K

into linear factors.

g

⁢

(

x

)

=

x

4

-

2

=

(

x

2

-

2

)

⋅

(

x

2

+

2

)

=

(

x

-

2

1

/

4

)

⋅

(

x

+

2

1

/

4

)

⋅

(

x

2

+

2

)

The

Galois closure

of

K

over

ℚ

is

L

=

ℚ

⁢

(

2

1

/

4

,

i

)

.

Title

example of

normal extension

Canonical name

ExampleOfNormalExtension

Date of creation

2013-03-22 14:30:46

Last modified on

2013-03-22 14:30:46

Owner

alozano (2414)

Last modified by

alozano (2414)

Numerical id

4

Author

alozano (2414)

Entry type

Example

Classification

msc 12F10

Related topic

GaloisExtension

Related topic

CompositumOfAGaloisExtensionAndAnotherExtensionIsGalois

Related topic

NormalIsNotTransitive

Related topic

GaloisIsNotTransitive

Generated on Fri Feb 9 16:43:20 2018 by

LaTeXML
fixed field

fixed field

Let

K

/

F

be a

field extension

with

Galois group

G

=

Gal

⁡

(

K

/

F

)

, and let

H

be a

subgroup

of

G

. The

fixed field

of

H

in

K

is the set

K

H

:=

{

x

∈

K

∣

σ

⁢

(

x

)

=

x

⁢

for all

⁢

σ

∈

H

}

.

The set

K

H

is always a field, and

F

⊂

K

H

⊂

K

.

Title

fixed field

Canonical name

FixedField

Date of creation

2013-03-22 12:08:24

Last modified on

2013-03-22 12:08:24

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 16:43:27 2018 by

LaTeXML
Galois closure

Galois closure

Let

K

be an

extension field

of

F

. A

Galois closure

of

K

/

F

is a field

L

⊇

K

that is a

Galois extension

of

F

and is minimal in that respect, i.e. no

proper subfield

of

L

containing

K

is normal over

F

.

Title

Galois closure

Canonical name

GaloisClosure

Date of creation

2013-03-22 13:09:34

Last modified on

2013-03-22 13:09:34

Owner

scanez (1021)

Last modified by

scanez (1021)

Numerical id

6

Author

scanez (1021)

Entry type

Definition

Classification

msc 12F10

Generated on Fri Feb 9 16:43:28 2018 by

LaTeXML
Galois conjugate

Galois conjugate

Let

K

be a field, and let

L

be a

separable closure

. For any

x

∈

K

, the

Galois conjugates

of

x

are the elements of

L

which are in the orbit of

x

under the

group action

of the

absolute Galois group

G

K

on

L

.

Title

Galois conjugate

Canonical name

GaloisConjugate

Date of creation

2013-03-22 12:12:25

Last modified on

2013-03-22 12:12:25

Owner

djao (24)

Last modified by

djao (24)

Numerical id

7

Author

djao (24)

Entry type

Definition

Classification

msc 12F10

Generated on Fri Feb 9 16:43:30 2018 by

LaTeXML
Galois extension

Galois extension

A

field extension

is

Galois

if it is normal and

separable

.

Title

Galois extension

Canonical name

GaloisExtension

Date of creation

2013-03-22 12:08:27

Last modified on

2013-03-22 12:08:27

Owner

djao (24)

Last modified by

djao (24)

Numerical id

7

Author

djao (24)

Entry type

Definition

Classification

msc 12F10

Generated on Fri Feb 9 16:43:31 2018 by

LaTeXML
Galois group

Galois group

The

Galois group

Gal

⁡

(

K

/

F

)

of a

field extension

K

/

F

is the group of all

field automorphisms

σ

:

K

→

K

of

K

which fix

F

(i.e.,

σ

⁢

(

x

)

=

x

for all

x

∈

F

). The

group operation

is given by

composition

: for two

automorphisms

σ

1

,

σ

2

∈

Gal

⁡

(

K

/

F

)

, given by

σ

1

:

K

→

K

and

σ

2

:

K

→

K

, the

product

σ

1

⋅

σ

2

∈

Gal

⁡

(

K

/

F

)

is the composite of the two maps

σ

1

∘

σ

2

:

K

→

K

.

The

Galois group

of a

polynomial

f

⁢

(

x

)

∈

F

⁢

[

x

]

is defined to be the Galois group of the

splitting field

of

f

⁢

(

x

)

over

F

.

Title

Galois group

Canonical name

GaloisGroup

Date of creation

2013-03-22 12:08:19

Last modified on

2013-03-22 12:08:19

Owner

djao (24)

Last modified by

djao (24)

Numerical id

8

Author

djao (24)

Entry type

Definition

Classification

msc 12F10

Related topic

FundamentalTheoremOfGaloisTheory

Related topic

InfiniteGaloisTheory

Generated on Fri Feb 9 16:43:33 2018 by

LaTeXML
Galois is not transitive

Galois is not transitive

The phrase “

Galois is not transitive

” is a mnemonic for the statement “The

relation

‘is a

Galois extension

of’ is not

transitive

.” This means that, if

K

/

F

and

L

/

K

are Galois extensions (

http://planetmath.org/GaloisExtension

), it does not follow that

L

/

F

is Galois. This follows immediately from the fact that

normal is not transitive

. See example of

normal extension

for more details.

Title

Galois is not transitive

Canonical name

GaloisIsNotTransitive

Date of creation

2013-03-22 16:00:31

Last modified on

2013-03-22 16:00:31

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

10

Author

Wkbj79 (1863)

Entry type

Definition

Classification

msc 12F10

Related topic

ExampleOfNormalExtension

Generated on Fri Feb 9 16:43:39 2018 by

LaTeXML
Galois subfields of real radical extensions are at most quadratic

Galois subfields of real radical extensions are at most quadratic

Theorem 1

.

Suppose

F

⊂

L

⊂

K

=

F

⁢

(

α

n

)

⊂

R

are fields with

α

∈

F

and

L

Galois over

F

. Then

[

L

:

F

]

≤

2

.

Proof.

Let

ζ

n

be a

primitive

n

th

root of unity

, and define

F

′

=

F

⁢

(

ζ

n

)

,

L

′

=

L

⁢

(

ζ

n

)

, and

K

′

=

K

⁢

(

ζ

n

)

=

F

′

⁢

(

α

n

)

.

\xymatrix

⁢

@

⁢

R

⁢

1

⁢

p

⁢

c

⁢

@

⁢

C

⁢

.3

⁢

p

⁢

c

⁢

&

⁢

K

′

=

K

⁢

(

ζ

n

)

=

F

′

⁢

(

α

n

)

⁢

\ar

⁢

@

-

[

d

⁢

l

]

⁢

\ar

⁢

@

-

[

d

⁢

r

]

⁢

&

⁢

&

⁢

K

=

F

⁢

(

α

n

)

⁢

\ar

⁢

@

-

[

d

⁢

r

]

⁢

&

⁢

&

⁢

L

′

=

L

⁢

(

ζ

n

)

⁢

\ar

⁢

@

-

[

d

⁢

l

]

⁢

\ar

⁢

@

-

[

d

⁢

r

]

⁢

&

⁢

&

⁢

L

⁢

\ar

⁢

@

-

[

d

⁢

r

]

⁢

&

⁢

&

⁢

F

′

=

F

⁢

(

ζ

n

)

⁢

\ar

⁢

@

-

[

d

⁢

l

]

⁢

&

⁢

&

⁢

F

⁢

&

Now,

L

′

/

F

′

is Galois since

L

/

F

is. But

K

′

is a

Kummer extension

of

F

′

, so has cyclic

Galois group

and thus

L

′

/

F

′

has cyclic Galois group as well (being a

quotient

of

Gal

⁡

(

K

′

/

F

′

)

). Thus

L

′

is a Kummer extension of

F

′

, so that

L

′

=

F

′

⁢

(

β

n

)

for some

β

∈

F

′

. It follows that

L

=

F

⁢

(

β

n

)

. But since

L

is Galois over

F

, it follows that

n

≤

2

(since otherwise in order to be Galois,

L

would have to contain the non-real

n

th

roots of unity).

Title

Galois subfields of real radical extensions are at most quadratic

Canonical name

GaloisSubfieldsOfRealRadicalExtensionsAreAtMostQuadratic

Date of creation

2013-03-22 17:43:05

Last modified on

2013-03-22 17:43:05

Owner

rm50 (10146)

Last modified by

rm50 (10146)

Numerical id

7

Author

rm50 (10146)

Entry type

Theorem

Classification

msc 12F10

Classification

msc 12F05

Generated on Fri Feb 9 16:43:41 2018 by

LaTeXML
normal closure

normal closure

Let

K

be an

extension field

of

F

. A

normal closure

of

K

/

F

is a field

L

⊇

K

that is a

normal extension

of

F

and is

minimal

in that respect, i.e. no

proper subfield

of

L

containing

K

is normal over

F

. If

K

is an

algebraic extension

of

F

, then a normal closure for

K

/

F

exists and is unique up to

isomorphism

.

Title

normal closure

Canonical name

NormalClosure

Date of creation

2013-03-22 13:09:36

Last modified on

2013-03-22 13:09:36

Owner

scanez (1021)

Last modified by

scanez (1021)

Numerical id

5

Author

scanez (1021)

Entry type

Definition

Classification

msc 12F10

Generated on Fri Feb 9 16:43:45 2018 by

LaTeXML
normal extension

normal extension

A

field extension

K

/

F

is

normal

if every

irreducible polynomial

f

∈

F

⁢

[

x

]

which has at least one root in

K

splits (factors into a

product

of linear factors) in

K

⁢

[

x

]

.

An

extension

K

/

F

of finite degree is normal if and only if there exists a

polynomial

p

∈

F

⁢

[

x

]

such that

K

is the

splitting field

for

p

over

F

.

Title

normal extension

Canonical name

NormalExtension

Date of creation

2013-03-22 12:08:15

Last modified on

2013-03-22 12:08:15

Owner

djao (24)

Last modified by

djao (24)

Numerical id

10

Author

djao (24)

Entry type

Definition

Classification

msc 12F10

Synonym

normal

Related topic

SplittingField

Generated on Fri Feb 9 16:43:46 2018 by

LaTeXML