aarjaneiro/planetmath huggingface dataset:
proof that the compositum of a Galois extension and another extension is Galois

proof that the compositum of a Galois extension and another extension is Galois

Proof.

The

diagram

of the situation of the

theorem

is:

\xymatrix

⁢

@

⁢

R

⁢

1

⁢

p

⁢

c

⁢

@

⁢

C

⁢

1

⁢

p

⁢

c

⁢

&

⁢

&

⁢

&

⁢

\ar

⁢

@

-

[

l

⁢

l

⁢

l

⁢

d

]

⁢

\ar

⁢

@

-

[

r

⁢

d

⁢

d

]

⁢

E

⁢

F

⁢

\ar

⁢

@

-

[

r

⁢

d

⁢

d

]

⁢

E

⁢

&

⁢

&

⁢

&

⁢

&

⁢

\ar

⁢

@

-

[

l

⁢

l

⁢

l

⁢

d

]

⁢

F

⁢

&

⁢

\ar

⁢

@

-

[

d

]

⁢

E

∩

F

⁢

&

⁢

K

To see that

E

⁢

F

/

F

is Galois, note that since

E

/

K

is Galois,

E

is a

splitting field

of a set of

polynomials

over

K

; clearly

E

⁢

F

is a splitting field of the same set of polynomials over

F

. Also, if

f

∈

K

⁢

[

x

]

is

separable

over

K

, then also

f

is separable over

F

. Thus

E

⁢

F

is normal and separable over

F

, so is Galois.

E

is obviously Galois over

E

∩

F

since

E

∩

F

⊃

K

.

Let

r

be the

restriction

map

r

:

H

=

Gal

⁡

(

E

⁢

F

/

F

)

→

Gal

⁡

(

E

/

K

)

:

σ

↦

σ

|

E

r

is clearly a

group homomorphism

, and since

E

is normal over

K

,

r

is well-defined.

Claim

r

is

injective

. For suppose

σ

∈

Gal

⁡

(

E

⁢

F

/

F

)

and

σ

|

E

is the

identity

. Then

σ

is fixed on

F

(since it is in

Gal

⁡

(

E

⁢

F

/

F

)

and on

E

(since its restriction to

E

is the identity), so is fixed on

E

⁢

F

and thus is itself the identity.

Now, the image of

r

is a

subgroup

of

Gal

⁡

(

E

/

K

)

with

fixed field

L

, and thus the image of

r

is

Gal

⁡

(

E

/

L

)

. Claim

E

∩

F

=

L

.

⊂

is obvious: any

element

x

∈

E

∩

F

is fixed by

σ

|

E

for each

σ

∈

H

since

σ

fixes

F

. Thus

E

∩

F

⊂

L

. To see the reverse

inclusion

, choose

x

∈

L

; then

x

is fixed by each

r

⁢

(

σ

)

for

σ

∈

H

. But

x

∈

L

⊂

E

, so that (as an element of

E

),

x

is fixed by each

σ

∈

H

. Thus

x

∈

F

so that

x

∈

E

∩

F

.

Thus

L

=

E

∩

F

, and

r

is then an

isomorphism

Gal

⁡

(

E

⁢

F

/

F

)

≅

Gal

⁡

(

E

/

E

∩

F

)

.
∎

References

1

Morandi, P.,

Field and

Galois Theory

, Springer, 1996.

Title

proof that the

compositum

of a

Galois extension

and another

extension

is Galois

Canonical name

ProofThatTheCompositumOfAGaloisExtensionAndAnotherExtensionIsGalois

Date of creation

2013-03-22 18:41:58

Last modified on

2013-03-22 18:41:58

Owner

rm50 (10146)

Last modified by

rm50 (10146)

Numerical id

6

Author

rm50 (10146)

Entry type

Proof

Classification

msc 11R32

Classification

msc 12F99

Generated on Fri Feb 9 15:44:53 2018 by

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Shafarevich’s theorem

Shafarevich’s theorem

Theorem

(Shafarevich’s Theorem).

Every finite

solvable group

occurs as a

Galois group

of a

finite extension

of

Q

.

One can find a proof in

[

S-W

]

.

References

S-W

Alexander Schmidt and Kay Wingberg,

http://www.math.uiuc.edu/Algebraic-Number-Theory/0136/

Shafarevich’s Theorem on Solvable Groups as Galois Groups.

Title

Shafarevich’s theorem

Canonical name

ShafarevichsTheorem

Date of creation

2013-03-22 15:01:17

Last modified on

2013-03-22 15:01:17

Owner

alozano (2414)

Last modified by

alozano (2414)

Numerical id

8

Author

alozano (2414)

Entry type

Theorem

Classification

msc 11R32

Related topic

InverseGaloisProblem

Related topic

Solvable

Generated on Fri Feb 9 15:44:55 2018 by

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p-adic cyclotomic character

p

-adic cyclotomic character

Let

G

ℚ

=

Gal

⁡

(

ℚ

¯

/

ℚ

)

be the absolute Galois group of

ℚ

. The purpose of this entry is to define, for every prime

p

, a

Galois representation

:

χ

p

:

G

ℚ

⟶

ℤ

p

×

where

ℤ

p

×

is the

group of units

of

ℤ

p

, the

p

-adic

integers

.

χ

p

is a

ℤ

p

×

valued

character

, usually called the

cyclotomic character

of

G

ℚ

, or the

p

-adic cyclotomic Galois representation of

G

ℚ

. Here is the construction:

For each

n

≥

1

, let

ζ

p

n

be a

primitive

p

n

-th

root of unity

and let

K

n

=

ℚ

⁢

(

ζ

p

n

)

be the corresponding

cyclotomic extension

of

ℚ

. By the basic theory of cyclotomic extensions, we know that

Gal

⁡

(

K

n

/

ℚ

)

≅

(

ℤ

/

p

n

⁢

ℤ

)

×

.

Moreover, the restriction map

Gal

⁡

(

K

n

+

1

/

ℚ

)

→

Gal

⁡

(

K

n

/

ℚ

)

is given by

reduction

modulo

p

n

from

(

ℤ

/

p

n

+

1

⁢

ℤ

)

×

to

(

ℤ

/

p

n

⁢

ℤ

)

×

.

Therefore, for each

n

we can

construct

a

representation

:

χ

p

,

n

:

G

ℚ

→

Gal

⁡

(

K

n

/

ℚ

)

→

(

ℤ

/

p

n

⁢

ℤ

)

×

where the first map is simply restriction to

K

n

and the second map is an

isomorphism

. By the remarks above, the representations

χ

p

,

n

are coherent in a strong sense, i.e.

χ

p

,

n

+

1

⁢

(

σ

)

≡

χ

p

,

n

⁢

(

σ

)

mod

p

n

.

Therefore, one can construct a “big” Galois representation:

χ

p

:

G

ℚ

⟶

ℤ

p

×

by requiring

χ

⁢

(

σ

)

≡

χ

p

,

n

⁢

(

σ

)

mod

p

n

, for every

n

≥

1

.

One can rephrase the above

definition

as follows. Let

σ

∈

G

ℚ

. We need to define a

group homomorphism

χ

p

:

G

ℚ

→

ℤ

p

×

, so we need to first define

χ

p

⁢

(

σ

)

and then check that it is a

homomorphism

. By the theory,

σ

⁢

(

ζ

p

n

)

is another primitive

p

n

-th root of unity, thus

σ

⁢

(

ζ

p

n

)

=

ζ

p

n

t

n

for some integer

1

≤

t

n

≤

p

n

-

1

with

gcd

⁡

(

t

n

,

p

)

=

1

(so

t

n

is a unit modulo

p

n

). Moreover,

σ

⁢

(

ζ

p

n

-

1

)

=

σ

⁢

(

ζ

p

n

p

)

=

ζ

p

n

p

⁢

t

n

=

ζ

p

n

-

1

t

n

Therefore,

t

n

≡

t

n

-

1

modulo

p

n

-

1

. Thus, we may define:

χ

p

⁢

(

σ

)

=

lim

←

⁡

t

n

∈

ℤ

p

and as we have shown,

χ

p

⁢

(

σ

)

is a unit of

ℤ

p

. Finally, the reader should check that

χ

p

is a group homomorphism.

Title

p

-adic cyclotomic character

Canonical

name

PadicCyclotomicCharacter

Date of creation

2013-03-22 15:36:16

Last modified on

2013-03-22 15:36:16

Owner

alozano (2414)

Last modified by

alozano (2414)

Numerical id

6

Author

alozano (2414)

Entry type

Definition

Classification

msc 11R34

Classification

msc 11R32

Classification

msc 11R04

Synonym

p

-adic cyclotomic Galois representation

Synonym

cyclotomic character

Generated on Fri Feb 9 15:44:58 2018 by

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Artin map

Artin map

Let

L

/

K

be a

Galois extension

of

number fields

, with

rings of integers

𝒪

L

and

𝒪

K

. For any

finite prime

𝔓

⊂

L

lying over a prime

𝔭

∈

K

, let

D

⁢

(

𝔓

)

denote the

decomposition group

of

𝔓

, let

T

⁢

(

𝔓

)

denote the inertia group of

𝔓

, and let

l

:=

𝒪

L

/

𝔓

and

k

:=

𝒪

K

/

𝔭

be the

residue fields

. The

exact sequence

1

⟶

T

⁢

(

𝔓

)

⟶

D

⁢

(

𝔓

)

⟶

Gal

⁡

(

l

/

k

)

⟶

1

yields an

isomorphism

D

⁢

(

𝔓

)

/

T

⁢

(

𝔓

)

≅

Gal

⁡

(

l

/

k

)

. In particular, there is a unique

element

in

D

⁢

(

𝔓

)

/

T

⁢

(

𝔓

)

, denoted

[

L

/

K

,

𝔓

]

, which maps to the

q

th

power

Frobenius map

Frob

q

∈

Gal

⁡

(

l

/

k

)

under this isomorphism (where

q

is the number of elements in

k

). The notation

[

L

/

K

,

𝔓

]

is referred to as the

Artin symbol

of the

extension

L

/

K

at

𝔓

.

If we add the additional

assumption

that

𝔭

is

unramified

, then

T

⁢

(

𝔓

)

is the

trivial group

, and

[

L

/

K

,

𝔓

]

in this situation is an element of

D

⁢

(

𝔓

)

⊂

Gal

⁡

(

L

/

K

)

, called the

Frobenius automorphism

of

𝔓

.

If, furthermore,

L

/

K

is an

abelian extension

(that is,

Gal

⁡

(

L

/

K

)

is an

abelian group

), then

[

L

/

K

,

𝔓

]

=

[

L

/

K

,

𝔓

′

]

for any other prime

𝔓

′

⊂

L

lying over

𝔭

. In this case, the Frobenius automorphism

[

L

/

K

,

𝔓

]

is denoted

(

L

/

K

,

𝔭

)

; the change in notation from

𝔓

to

𝔭

reflects the fact that the

automorphism

is determined by

𝔭

∈

K

independent of which prime

𝔓

of

L

above it is chosen for use in the above construction.

Definition 1

.

Let

S

be a

finite set

of primes of

K

, containing all the primes that ramify in

L

. Let

I

K

S

denote the

subgroup

of the group

I

K

of

fractional ideals

of

K

which is generated by all the primes in

K

that are not in

S

. The

Artin map

ϕ

L

/

K

:

I

K

S

⟶

Gal

⁡

(

L

/

K

)

is the map given by

ϕ

L

/

K

⁢

(

𝔭

)

:=

(

L

/

K

,

𝔭

)

for all primes

𝔭

∉

S

, extended linearly to

I

K

S

.

Title

Artin map

Canonical name

ArtinMap

Date of creation

2013-03-22 12:34:55

Last modified on

2013-03-22 12:34:55

Owner

djao (24)

Last modified by

djao (24)

Numerical id

9

Author

djao (24)

Entry type

Definition

Classification

msc 11R37

Related topic

RayClassField

Defines

Artin symbol

Defines

Frobenius automorphism

Generated on Fri Feb 9 15:45:00 2018 by

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Dedekind domain

Dedekind domain

A

Dedekind domain

is a

commutative

integral domain

R

for which:

•

Every ideal in

R

is

finitely generated

.

•

Every nonzero

prime ideal

is a

maximal ideal

.

•

The domain

R

is

integrally closed

in its

field of fractions

.

It is worth noting that the second clause above implies that the

maximal

length of a strictly increasing chain of prime ideals is 1, so the

Krull dimension

of any Dedekind domain is at most 1. In particular, the affine ring of an

algebraic set

is a Dedekind domain if and only if the set is normal,

irreducible

, and 1-dimensional.

Every Dedekind domain is a

noetherian ring

.

If

K

is a

number field

, then

𝒪

K

, the

ring of algebraic integers

of

K

, is a Dedekind domain.

Title

Dedekind domain

Canonical name

DedekindDomain

Date of creation

2013-03-22 12:36:06

Last modified on

2013-03-22 12:36:06

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

16

Author

mathcam (2727)

Entry type

Definition

Classification

msc 11R37

Classification

msc 11R04

Related topic

IntegralClosure

Related topic

PruferDomain

Related topic

MultiplicationRing

Related topic

PrimeIdealFactorizationIsUnique

Related topic

EquivalentCharacterizationsOfDedekindDomains

Related topic

ProofThatADomainIsDedekindIfItsIdealsAreInvertible

Related topic

ProofThatADomainIsDedekindIfItsIdealsAreProductsOfPrimes

Related topic

ProofThatADomai

Generated on Fri Feb 9 15:45:05 2018 by

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ideal decomposition in Dedekind domain

ideal decomposition in Dedekind domain

According to the entry “

fractional ideal

(

http://planetmath.org/FractionalIdeal

)”, we can that in a

Dedekind domain

R

, each non-zero integral ideal

𝔞

may be written as a product of finitely many

prime ideals

𝔭

i

of

R

,

𝔞

=

𝔭

1

⁢

𝔭

2

⁢

…

⁢

𝔭

k

.

The product decomposition is unique up to the order of the factors.  This is stated and proved, with more general assumptions, in the entry “prime ideal factorisation is unique (

http://planetmath.org/PrimeIdealFactorizationIsUnique

)”.

Corollary.

If

α

1

,

α

2

, …,

α

m

are elements of a Dedekind domain

R

and

n

is a

positive integer

, then one has

(

α

1

,

α

2

,

…

,

α

m

)

n

=

(

α

1

n

,

α

2

n

,

…

,

α

m

n

)

(1)

for the ideals of

R

.

This corollary may be proven by induction on the number

m

of the

n

).

Title

ideal decomposition in Dedekind domain

Canonical name

IdealDecompositionInDedekindDomain

Date of creation

2015-05-05 19:05:43

Last modified on

2015-05-05 19:05:43

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

13

Author

pahio (2872)

Entry type

Topic

Classification

msc 11R37

Classification

msc 11R04

Related topic

ProductOfFinitelyGeneratedIdeals

Related topic

PolynomialCongruence

Related topic

CancellationIdeal

Related topic

DivisibilityInRings

Related topic

IdealsOfADiscreteValuationRingArePowersOfItsMaximalIdeal

Related topic

DivisorTheory

Related topic

GreatestCommonDivisorOfSeveralIntegers

Generated on Fri Feb 9 15:45:07 2018 by

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ideals in a Dedekind domain

ideals in a Dedekind domain

Let

R

be a

Dedekind domain

,
and let

𝔞

and

𝔟

be ideals of

R

.
Then there is an element

ω

and an ideal

𝔠

of

R

such that

𝔞

⁢

𝔠

=

(

ω

)

and

𝔟

+

𝔠

=

R

.

This result was proved by Steinitz in 1911.

Title

ideals in a Dedekind domain

Canonical name

IdealsInADedekindDomain

Date of creation

2013-03-22 12:49:50

Last modified on

2013-03-22 12:49:50

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

9

Author

yark (2760)

Entry type

Theorem

Classification

msc 11R37

Classification

msc 11R04

Related topic

DivisorAsFactorOfPrincipalDivisor

Related topic

FundamentalTheoremOfIdealTheory

Generated on Fri Feb 9 15:45:08 2018 by

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Kronecker’s Jugendtraum

Kronecker’s Jugendtraum

Kronecker’s

Jugendtraum

(Jugendtraum is German for “youthful dream”) describes a central problem in

class field theory

, to explicitly describe the

abelian extensions

of an arbitrary

number field

K

in of values of

transcendental functions

.

Class field theory gives a solution to this problem in the case where

K

=

ℚ

, the

field of rational

numbers. Specifically, the

Kronecker-Weber theorem

gives that any number field sits inside one of the

cyclotomic fields

ℚ

⁢

(

ζ

n

)

for some

n

. Refining this only slightly gives that we can explicitly generate all abelian extensions of

ℚ

by adjoining values of the transcendental function

e

2

⁢

π

⁢

i

⁢

z

for certain points

z

∈

ℚ

/

ℤ

.

A slightly more complicated example is when

K

is a quadratic

imaginary

extension

of

ℚ

, in which case Kronecker’s Jugendtraum has been solved by the theory of “

complex multiplication

” (see CM-field). The specific transcendental functions which generate all these abelian extensions are the

j

-

function

(as in

elliptic curves

) and Weber’s

w

-function.

Though there are partial results in the cases of CM-fields or

real quadratic fields

, the problem is largely still open (

http://planetmath.org/OpenQuestion

), and earned great prestige by being included as Hilbert’s twelfth problem.

Title

Kronecker’s Jugendtraum

Canonical name

KroneckersJugendtraum

Date of creation

2013-03-22 15:01:08

Last modified on

2013-03-22 15:01:08

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

7

Author

mathcam (2727)

Entry type

Definition

Classification

msc 11R37

Generated on Fri Feb 9 15:45:10 2018 by

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modulus

modulus

A

modulus

for a

number field

K

is a formal product

∏

𝔭

𝔭

n

𝔭

where

•

The product is taken over all

finite primes

and infinite primes of

K

•

The exponents

n

𝔭

are

nonnegative integers

•

All but finitely many of the

n

𝔭

are zero

•

For every real prime

𝔭

, the exponent

n

𝔭

is either 0 or 1

•

For every complex prime

𝔭

, the exponent

n

𝔭

is 0

A modulus can be written as a product of its finite part

∏

𝔭

⁢

finite

𝔭

n

𝔭

and its

infinite

part

∏

𝔭

⁢

real

𝔭

n

𝔭

,

with the finite part equal to some ideal in the

ring of integers

𝒪

K

of

K

, and the infinite part equal to the product of some subcollection of the real primes of

K

.

Title

modulus

Canonical name

Modulus

Date of creation

2013-03-22 12:35:26

Last modified on

2013-03-22 12:35:26

Owner

djao (24)

Last modified by

djao (24)

Numerical id

4

Author

djao (24)

Entry type

Definition

Classification

msc 11R37

Generated on Fri Feb 9 15:45:12 2018 by

LaTeXML
multiplicative congruence

multiplicative congruence

Let

𝔭

be any

real prime

of a

number field

K

, and write

i

:

K

⟶

ℝ

for the corresponding

real embedding

of

K

. We say two elements

α

,

β

∈

K

are

multiplicatively congruent

mod

𝔭

if the

real numbers

i

⁢

(

α

)

and

i

⁢

(

β

)

are either both

positive

or both negative.

Now let

𝔭

be a finite prime of

K

, and write

(

𝒪

K

)

𝔭

for the

localization

of the

ring of integers

𝒪

K

of

K

at

𝔭

. For any natural number

n

, we say

α

and

β

are

multiplicatively congruent

mod

𝔭

n

if they are members of the same coset of the

subgroup

1

+

𝔭

n

⁢

(

𝒪

K

)

𝔭

of the

multiplicative group

K

×

of

K

.

If

𝔪

is any

modulus

for

K

, with factorization

𝔪

=

∏

𝔭

𝔭

n

𝔭

,

then we say

α

and

β

are

multiplicatively congruent

mod

𝔪

if they are multiplicatively congruent mod

𝔭

n

𝔭

for every prime

𝔭

appearing in the factorization of

𝔪

.

Multiplicative congruence of

α

and

β

mod

𝔪

is commonly denoted using the notation

α

≡

*

β

(

mod

𝔪

)

.

Title

multiplicative congruence

Canonical

name

MultiplicativeCongruence

Date of creation

2013-03-22 12:50:16

Last modified on

2013-03-22 12:50:16

Owner

djao (24)

Last modified by

djao (24)

Numerical id

4

Author

djao (24)

Entry type

Definition

Classification

msc 11R37

Synonym

multiplicatively congruent

Related topic

Congruence2

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push-down theorem on class numbers

push-down theorem on class numbers

As in the parent (

http://planetmath.org/ClassNumberDivisibilityInExtensions

) entry, given a

number field

K

, the

class number

of

K

is denoted by

h

K

.

Theorem

(Pushing-Down Theorem).

Let

E

/

F

be a

p

-

extension

of number fields and suppose that only one

prime ideal

of

F

is

ramified

in

E

and that this prime is

totally ramified

. Then

p

|

h

E

implies

p

|

h

F

.

References

Fröh

A. Fröhlich,

On a method for the determination of class number factors in number fields

, Mathematika, 4 (1957), 113-121.

Iwas

K. Iwasawa,

A note on Class Numbers of

Algebraic Number Fields

, Abh. Math. Sem. Univ. Hamburg, 20 (1956), 257-258.

Title

push-down theorem on class numbers

Canonical name

PushdownTheoremOnClassNumbers

Date of creation

2013-03-22 15:05:19

Last modified on

2013-03-22 15:05:19

Owner

alozano (2414)

Last modified by

alozano (2414)

Numerical id

6

Author

alozano (2414)

Entry type

Theorem

Classification

msc 11R37

Classification

msc 11R32

Classification

msc 11R29

Related topic

IdealClass

Related topic

PExtension

Related topic

ExtensionsWithoutUnramifiedSubextensionsAndClassNumberDivisibility

Related topic

ClassNumberDivisibilityInPExtensions

Related topic

ClassNumbersAndDiscriminantsTopicsOnClassGroups

Generated on Fri Feb 9 15:45:16 2018 by

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ray class field

ray class field

Proposition 1

.

Let

L

/

K

be a finite

abelian extension

of

number fields

, and let

O

K

be the

ring of integers

of

K

. There exists an

integral ideal

C

⊂

O

K

,

divisible

by
precisely the

prime ideals

of

K

that ramify in

L

, such that

(

(

α

)

,

L

/

K

)

=

1

,

∀

α

∈

K

∗

,

α

≡

1

⁢

mod

⁡

𝒞

where

(

(

α

)

,

L

/

K

)

is the

Artin map

.

Definition 1

.

The

conductor

of a finite abelian extension

L

/

K

is the
largest ideal

C

L

/

K

⊂

O

K

satisfying
the above

properties

.

Note that there is a “largest ideal” with this condition because
if

proposition

1 is true for

𝒞

1

,

𝒞

2

then it
is also true for

𝒞

1

+

𝒞

2

.

Definition 2

.

Let

I

be an integral ideal of

K

. A

ray class
field

of

K

(modulo

I

) is a finite abelian extension

K

I

/

K

with the property that for any other finite
abelian extension

L

/

K

with conductor

C

L

/

K

,

𝒞

L

/

K

∣

ℐ

⇒

L

⊂

K

ℐ

Note: It can be proved that there is a unique ray class field with
a given conductor. In words, the ray class field is the biggest
abelian extension of

K

with a given conductor (although the
conductor of

K

ℐ

does not necessarily equal

ℐ

!, see example

2

).

Remark

: Let

𝔭

be a prime of

K

unramified

in

L

, and let

𝔓

be a prime above

𝔭

. Then

(

𝔭

,

L

/

K

)

=

1

if and only if the

extension

of

residue fields

is of degree 1

[

𝒪

L

/

𝔓

:

𝒪

K

/

𝔭

]

=

1

if and only if

𝔭

splits completely in

L

. Thus we obtain a

characterization

of the ray class field of conductor

𝒞

as the abelian extension of

K

such that a prime of

K

splits completely if and only if it is of the form

(

α

)

,

α

∈

K

∗

,

α

≡

1

⁢

mod

⁡

𝒞

Examples

:

1.

The ray class field of

ℚ

of conductor

N

⁢

ℤ

is the

N

t

⁢

h

-

cyclotomic extension

of

ℚ

. More concretely, let

ζ

N

be a primitive

N

t

⁢

h

root of unity

. Then

ℚ

N

⁢

ℤ

=

ℚ

⁢

(

ζ

N

)

2.

ℚ

⁢

(

i

)

(

2

)

=

ℚ

⁢

(

i

)

so the conductor of

ℚ

⁢

(

i

)

(

2

)

/

ℚ

is

(

1

)

.

3.

K

(

1

)

, the ray class field of conductor

(

1

)

, is the
maximal abelian extension of

K

which is unramified everywhere.
It is, in fact, the

Hilbert class field

of

K

.

References

1

Artin/Tate,

Class Field Theory

. W.A.Benjamin Inc., New York.

Title

ray class field

Canonical

name

RayClassField

Date of creation

2013-03-22 13:54:01

Last modified on

2013-03-22 13:54:01

Owner

alozano (2414)

Last modified by

alozano (2414)

Numerical id

5

Author

alozano (2414)

Entry type

Definition

Classification

msc 11R37

Synonym

conductor

Related topic

ArtinMap

Related topic

ExistenceOfHilbertClassField

Related topic

NumberField

Related topic

AnExactSequenceForRayClassGroups

Defines

conductor of an extension

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Tchebotarev density theorem

Tchebotarev density theorem

Let

L

/

K

be any finite

Galois extension

of

number fields

with

Galois group

G

. For any

conjugacy class

C

⊂

G

, the subset of prime ideals

𝔭

⊂

K

which are

unramified

in

L

and satisfy the property

[

L

/

K

,

𝔓

]

∈

C

⁢

for any prime

⁢

𝔓

⊂

L

⁢

containing

⁢

𝔭

has analytic density

|

C

|

|

G

|

, where

[

L

/

K

,

𝔓

]

denotes the

Artin symbol

at

𝔓

.

Note that the conjugacy class of

[

L

/

K

,

𝔓

]

is

independent

of the choice of prime

𝔓

lying over

𝔭

, since any two such choices of primes are related by a Galois

automorphism

and their corresponding Artin symbols are

conjugate

by this same automorphism.

Title

Tchebotarev density theorem

Canonical name

TchebotarevDensityTheorem

Date of creation

2013-03-22 12:46:49

Last modified on

2013-03-22 12:46:49

Owner

djao (24)

Last modified by

djao (24)

Numerical id

5

Author

djao (24)

Entry type

Theorem

Classification

msc 11R37

Classification

msc 11R44

Classification

msc 11R45

Synonym

Chebotarev density theorem

Generated on Fri Feb 9 15:45:21 2018 by

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