aarjaneiro/planetmath huggingface dataset:
class number divisibility in p-extensions

class number divisibility in

p

-extensions

In this entry, the

class number

of a

number field

F

is denoted by

h

F

.

Theorem 1

.

Let

p

be a fixed

prime number

.

•

Let

F

/

K

be a

Galois extension

with

Galois group

Gal

⁡

(

F

/

K

)

and suppose

F

/

K

is a

p

-

extension

(so

Gal

⁡

(

F

/

K

)

is a

p

-group). Assume that there is at most one prime or

archimedean place

which ramifies in

F

/

K

. If

h

F

is

divisible

by

p

then

h

K

is also divisible by

p

.

•

Let

F

/

ℚ

be a Galois extension of the

rational numbers

and assume that

Gal

⁡

(

F

/

ℚ

)

is a

p

-group and at most one place (finite or

infinite

) ramifies then

h

F

is not divisible by

p

.

Title

class number

divisibility

in

p

-extensions

Canonical name

ClassNumberDivisibilityInPextensions

Date of creation

2013-03-22 15:07:38

Last modified on

2013-03-22 15:07:38

Owner

alozano (2414)

Last modified by

alozano (2414)

Numerical id

6

Author

alozano (2414)

Entry type

Theorem

Classification

msc 11R29

Classification

msc 11R37

Related topic

PushDownTheoremOnClassNumbers

Related topic

IdealClass

Related topic

PExtension

Related topic

ClassNumbersAndDiscriminantsTopicsOnClassGroups

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

LaTeXML
class number formula

class number formula

Let

K

be a

number field

with

[

K

:

ℚ

]

=

n

=

r

1

+

2

r

2

, where

r

1

denotes the number of

real embeddings

of

K

, and

2

⁢

r

2

is the
number of complex embeddings of

K

. Let

ζ

K

⁢

(

s

)

be the

Dedekind zeta function

of

K

. Also define the following

invariants

:

1.

h

K

is the

class number

, the number of

elements

in the
ideal class group of

K

.

2.

Reg

K

is the

regulator

of

K

.

3.

ω

K

is the number of

roots of unity

contained

in

K

.

4.

D

K

is the

discriminant

of the

extension

K

/

ℚ

.

Then:

Theorem 1

(Class Number Formula).

The Dedekind zeta function of

K

,

ζ

K

⁢

(

s

)

converges
absolutely for

ℜ

⁡

(

s

)

>

1

and extends to a

meromorphic function

defined for

ℜ

⁡

(

s

)

>

1

-

1

n

with only one

simple pole

at

s

=

1

. Moreover:

lim

s

→

1

⁡

(

s

-

1

)

⁢

ζ

K

⁢

(

s

)

=

2

r

1

⋅

(

2

⁢

π

)

r

2

⋅

h

K

⋅

Reg

K

ω

K

⋅

∣

D

K

∣

Note: This is the most general “

class number formula

”. In
particular cases, for example when

K

is a

cyclotomic extension

of

ℚ

, there are particular and more refined class number formulas.

Title

class number formula

Canonical name

ClassNumberFormula

Date of creation

2013-03-22 13:54:37

Last modified on

2013-03-22 13:54:37

Owner

alozano (2414)

Last modified by

alozano (2414)

Numerical id

5

Author

alozano (2414)

Entry type

Theorem

Classification

msc 11R29

Classification

msc 11R42

Related topic

FunctionalEquationOfTheRiemannZetaFunction

Related topic

DedekindZetaFunction

Related topic

IdealClass

Related topic

Regulator

Related topic

Discriminant

Related topic

NumberField

Related topic

ClassNumbersAndDiscriminantsTopicsOnClassGroups

Defines

class number formula

Generated on Fri Feb 9 15:43:45 2018 by

LaTeXML
discriminant

discriminant

1

Definitions

Let

R

be any

Dedekind domain

with

field of fractions

K

. Fix a finite dimensional

field extension

L

/

K

and let

S

denote the

integral closure

of

R

in

L

. For any basis

x

1

,

…

,

x

n

of

L

over

K

, the

determinant

Δ

⁢

(

x

1

,

…

,

x

n

)

:=

det

⁡

[

Tr

⁡

(

x

i

⁢

x

j

)

]

,

whose entries are the trace of

x

i

⁢

x

j

over all pairs

i

,

j

, is called the

discriminant

of the basis

x

1

,

…

,

x

n

. The ideal in

R

generated by all discriminants of the form

Δ

⁢

(

x

1

,

…

,

x

n

)

,

x

i

∈

S

is called the

discriminant ideal

of

S

over

R

, and denoted

Δ

⁢

(

S

/

R

)

.

In the special case where

S

is a free

R

–module, the discriminant ideal

Δ

⁢

(

S

/

R

)

is always a

principal ideal

, generated by any discriminant of the form

Δ

⁢

(

x

1

,

…

,

x

n

)

where

x

1

,

…

,

x

n

is a basis for

S

as an

R

–module. In particular, this situation holds whenever

K

and

L

are

number fields

.

2

Alternative notations

The discriminant is sometimes denoted with

disc

instead of

Δ

. In the context of number fields, one often writes

disc

⁡

(

L

/

K

)

for

disc

⁡

(

𝒪

L

/

𝒪

K

)

where

𝒪

L

and

𝒪

K

are the

rings of algebraic integers

of

L

and

K

. If

K

or

𝒪

K

is omitted, it is typically assumed to be

ℚ

or

ℤ

.

3

Properties

The discriminant is so named because it allows one to determine which ideals of

R

are

ramified

in

S

. Specifically, the

prime ideals

of

R

that ramify in

S

are precisely the ones that contain the discriminant ideal

Δ

⁢

(

S

/

R

)

. In the case

R

=

ℤ

, a

theorem

of Minkowski (

http://planetmath.org/MinkowskisConstant

) that any

ring of integers

S

of a number field larger than

ℚ

has discriminant

strictly

smaller than

ℤ

itself, and this fact combined with the previous result shows that any number field

K

≠

ℚ

admits at least one ramified prime over

ℚ

.

4

Other types of discriminants

In the special case where

L

=

K

⁢

[

x

]

is a primitive separable field extension of degree

n

, the discriminant

Δ

⁢

(

1

,

x

,

…

,

x

n

-

1

)

is equal to the polynomial discriminant (

http://planetmath.org/PolynomialDiscriminant

) of the

minimal polynomial

f

⁢

(

X

)

of

x

over

K

⁢

[

X

]

.

The discriminant of an

elliptic curve

can be obtained by taking the

polynomial

discrimiant of its Weierstrass polynomial, and the

modular discriminant

of a

complex

lattice

equals the discriminant of the elliptic curve represented by the corresponding lattice

quotient

.

Title

discriminant

Canonical

name

Discriminant1

Date of creation

2013-03-22 12:37:57

Last modified on

2013-03-22 12:37:57

Owner

djao (24)

Last modified by

djao (24)

Numerical id

12

Author

djao (24)

Entry type

Definition

Classification

msc 11R29

Related topic

IntegralBasis

Related topic

PolynomialDiscriminant

Related topic

ModularDiscriminant

Defines

discriminant ideal

Generated on Fri Feb 9 15:43:48 2018 by

LaTeXML
discriminant in algebraic number field

discriminant in algebraic number field

Let us consider the

elements

α

1

,

α

2

,

…

,

α

n

of an

algebraic number field

ℚ

⁢

(

ϑ

)

of degree (

http://planetmath.org/NumberField

)

n

.  Let

ϑ

1

=

ϑ

,

ϑ

2

,

…

,

ϑ

n

be the

algebraic conjugates

of the

primitive element

ϑ

and

α

i

=

r

i

(

ϑ

)

(

i

=

1

,

2

,

…

,

n

)

the

canonical forms

of the elements

α

i

.  Then the

ℚ

⁢

(

ϑ

)

-

conjugates

(

http://planetmath.org/CharacteristicPolynomialOfAlgebraicNumber

) of those elements are

α

i

(

j

)

=

r

i

⁢

(

ϑ

j

)

.

Using these, one can define the

discriminant

Δ

⁢

(

α

1

,

α

2

,

…

,

α

n

)

of the elenents

α

i

as

Δ

⁢

(

α

1

,

α

2

,

…

,

α

n

)

:=

det

⁡

(

r

i

⁢

(

ϑ

j

)

)

2

=

det

⁡

(

α

i

(

j

)

)

2

,

i.e.

Δ

⁢

(

α

1

,

α

2

,

…

,

α

n

)

:=

|

r

1

⁢

(

ϑ

1

)

r

1

⁢

(

ϑ

2

)

…

r

1

⁢

(

ϑ

n

)

r

2

⁢

(

ϑ

1

)

r

2

⁢

(

ϑ

2

)

…

r

2

⁢

(

ϑ

n

)

⋮

⋮

⋱

⋮

r

n

⁢

(

ϑ

1

)

r

n

⁢

(

ϑ

2

)

…

r

n

⁢

(

ϑ

n

)

|

2

=

|

α

1

(

1

)

α

1

(

2

)

…

α

1

(

n

)

α

2

(

1

)

α

2

(

2

)

…

α

2

(

n

)

⋮

⋮

⋱

⋮

α

n

(

1

)

α

n

(

2

)

…

α

n

(

n

)

|

2

.

Basing on the properties of

determinants

, one sees at once that the discriminant is of the numbers

α

i

.  The entry independence of

characteristic polynomial

on primitive element allows to see that the discriminant also does not depend on the used primitive element of the field.  Moreover, the method for multiplying the determinants enables to convert the discriminant into the form

Δ

⁢

(

α

1

,

α

2

,

…

,

α

n

)

=

|

S

⁢

(

α

1

⁢

α

1

)

S

⁢

(

α

1

⁢

α

2

)

…

S

⁢

(

α

1

⁢

α

n

)

S

⁢

(

α

2

⁢

α

1

)

S

⁢

(

α

2

⁢

α

2

)

…

S

⁢

(

α

2

⁢

α

n

)

⋮

⋮

⋱

⋮

S

⁢

(

α

n

⁢

α

1

)

S

⁢

(

α

n

⁢

α

2

)

…

S

⁢

(

α

n

⁢

α

n

)

|

,

where

S

is the trace

function

defined in

ℚ

⁢

(

ϑ

)

; therefore the discriminant is always a

rational number

(and an

integer

if every

α

i

is an

algebraic integer

of the field).  Cf. the parent entry (

http://planetmath.org/DiscriminantOfANumberField

).

Title

discriminant in algebraic number field

Canonical name

DiscriminantInAlgebraicNumberField

Date of creation

2013-03-22 19:09:23

Last modified on

2013-03-22 19:09:23

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

6

Author

pahio (2872)

Entry type

Definition

Classification

msc 11R29

Synonym

discriminant in terms of conjugates

Related topic

IndependenceOfCharacteristicPolynomialOnPrimitiveElement

Defines

discriminant

Generated on Fri Feb 9 15:43:51 2018 by

LaTeXML
discriminant of algebraic number

discriminant of algebraic number

Theorem.

If

ϑ

is an

algebraic number

of degree

n

with

minimal polynomial

f

⁢

(

x

)

, then the

of the number

ϑ

, i.e. the

discriminant

Δ

⁢

(

1

,

ϑ

,

…

,

ϑ

n

-

1

)

,  is

d

⁢

(

ϑ

)

=

(

-

1

)

n

⁢

(

n

-

1

)

2

⁢

N

⁢

(

f

′

⁢

(

ϑ

)

)

,

where N means the

absolute norm

.

Proof.

Let the

algebraic conjugates

of the number

ϑ

, i.e. all

complex

zeroes of

f

⁢

(

x

)

,  be

ϑ

1

=

ϑ

,

ϑ

2

,

…

,

ϑ

n

.  If

f

⁢

(

x

)

=

x

n

+

a

1

⁢

x

n

-

1

+

…

+

a

n

,  we have

f

′

⁢

(

ϑ

)

=

n

⁢

ϑ

n

-

1

+

(

n

-

1

)

⁢

a

1

⁢

ϑ

n

-

2

+

…

+

2

⁢

a

n

-

2

⁢

ϑ

+

a

n

-

1

∈

ℚ

⁢

(

ϑ

)

.

The norm (

http://planetmath.org/AbsoluteNorm

) of

f

′

⁢

(

ϑ

)

in

ℚ

⁢

(

ϑ

)

/

ℚ

is the product of all

http://planetmath.org/node/12046

ℚ

⁢

(

ϑ

)

-

conjugates

[

f

′

⁢

(

ϑ

)

]

(

i

)

of

f

′

⁢

(

ϑ

)

, which is

N

⁢

(

f

′

⁢

(

ϑ

)

)

=

[

f

′

⁢

(

ϑ

)

]

(

1

)

⁢

[

f

′

⁢

(

ϑ

)

]

(

2

)

⁢

⋯

⁢

[

f

′

⁢

(

ϑ

)

]

(

n

)

=

f

′

⁢

(

ϑ

1

)

⁢

f

′

⁢

(

ϑ

2

)

⁢

⋯

⁢

f

′

⁢

(

ϑ

n

)

.

On the other side, the polynonomial

f

⁢

(

x

)

in its linear factors is

f

⁢

(

x

)

=

(

x

-

ϑ

1

)

⁢

(

x

-

ϑ

2

)

⁢

⋯

⁢

(

x

-

ϑ

n

)

,

whence its derivative may be written

f

′

⁢

(

x

)

=

∑

ν

=

1

n

(

x

-

ϑ

1

)

⁢

⋯

⁢

(

x

-

ϑ

ν

-

1

)

⁢

(

x

-

ϑ

ν

+

1

)

⁢

⋯

⁢

(

x

-

ϑ

n

)

.

Substituting

x

=

ϑ

ν

gives simply

f

′

⁢

(

ϑ

ν

)

=

∏

j

≠

ν

(

ϑ

ν

-

ϑ

j

)

for

⁢

ν

=

1

,

…

,

n

.

Multiplying these equations we obtain

N

⁢

(

f

′

⁢

(

ϑ

)

)

=

∏

ν

=

1

n

f

′

⁢

(

ϑ

ν

)

=

∏

i

≠

j

(

ϑ

i

-

ϑ

j

)

.

The discriminant of

ϑ

is same as the discriminant of the equation

f

⁢

(

x

)

=

0

.  Therefore

d

⁢

(

ϑ

)

=

[

∏

i

<

j

(

ϑ

i

-

ϑ

j

)

]

2

,

where the number of the factors in the brackets is

(

n

-

1

)

+

(

n

-

2

)

+

…

+

1

=

(

n

-

1

)

⁢

n

2

.  Thus we obtain the asserted result

d

⁢

(

ϑ

)

=

[

∏

i

<

j

(

ϑ

i

-

ϑ

j

)

]

⋅

(

-

1

)

n

⁢

(

n

-

1

)

2

⁢

[

∏

j

<

i

(

ϑ

i

-

ϑ

j

)

]

=

(

-

1

)

n

⁢

(

n

-

1

)

2

⁢

∏

i

≠

j

(

ϑ

i

-

ϑ

j

)

=

(

-

1

)

n

⁢

(

n

-

1

)

2

⁢

N

⁢

(

f

′

⁢

(

ϑ

)

)

.

Title

discriminant of algebraic number

Canonical name

DiscriminantOfAlgebraicNumber

Date of creation

2013-03-22 17:49:59

Last modified on

2013-03-22 17:49:59

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

10

Author

pahio (2872)

Entry type

Theorem

Classification

msc 11R29

Related topic

Discriminant

Related topic

DerivativeOfPolynomial

Defines

discriminant of number

Generated on Fri Feb 9 15:43:54 2018 by

LaTeXML
example of a Bezout domain that is not a PID

example of a Bezout domain that is not a PID

Let

𝔸

be the ring of all

algebraic numbers

whose

minimal polynomials

are in

ℤ

⁢

[

x

]

; i.e. (

http://planetmath.org/Ie

), every

element of

𝔸

is an

algebraic integer

.

In the following example, ideals are considered to be of

𝔸

unless indicated otherwise via

intersection

with a

subring

of

𝔸

.

Let

I

be a ideal of

𝔸

. Then there exists a

positive integer

n

and

α

1

,

…

,

α

n

∈

𝔸

with

I

=

⟨

α

1

,

…

,

α

n

⟩

. Let

K

=

ℚ

⁢

(

α

1

,

…

,

α

n

)

, and let

𝒪

K

denote the

ring of integers

of

K

. Then

α

1

,

…

,

α

n

∈

𝒪

K

and

I

∩

𝒪

K

is an ideal of

𝒪

K

. Let

h

denote the

class number

of

K

. Then

(

I

∩

𝒪

K

)

h

=

⟨

β

⟩

∩

𝒪

K

for some

β

∈

𝒪

K

. Let

L

=

K

⁢

(

β

h

)

, and let

𝒪

L

denote the ring of integers of

L

. Then

(

I

∩

𝒪

L

)

h

=

[

(

I

∩

𝒪

K

)

⁢

𝒪

L

]

h

=

(

I

∩

𝒪

K

)

h

⁢

(

𝒪

L

)

h

=

(

⟨

β

⟩

∩

𝒪

K

)

⁢

𝒪

L

=

⟨

β

⟩

∩

𝒪

L

=

(

⟨

β

h

⟩

∩

𝒪

L

)

h

Since

unique factorization

of ideals holds in

𝒪

L

,

I

∩

𝒪

L

=

⟨

β

h

⟩

∩

𝒪

L

. Since

𝒪

K

⊆

𝒪

L

and

α

1

,

…

,

α

n

∈

I

∩

𝒪

K

⊆

I

∩

𝒪

L

=

⟨

β

h

⟩

∩

𝒪

L

, there exist

γ

1

,

…

,

γ

n

∈

𝒪

L

with

α

j

=

γ

j

⁢

β

h

for all positive integers

j

with

j

≤

n

. Thus,

I

=

⟨

α

1

,

…

,

α

n

⟩

=

⟨

γ

1

⁢

β

h

,

…

,

γ

n

⁢

β

h

⟩

⊆

⟨

β

h

⟩

. Since

I

⊆

⟨

β

h

⟩

and

I

∩

𝒪

L

=

⟨

β

h

⟩

∩

𝒪

L

,

I

=

⟨

β

h

⟩

. Hence,

I

is principal. It follows that

𝔸

is a

Bezout domain

.

On the other hand,

𝔸

is not a

principal ideal domain

(PID). For example, the ideal all of the

n

th roots (

http://planetmath.org/NthRoot

) of

2

,

J

=

⟨

2

,

2

,

2

3

,

…

⟩

, is an ideal of

𝔸

that is not principal.

Title

example of a Bezout domain that is not a PID

Canonical name

ExampleOfABezoutDomainThatIsNotAPID

Date of creation

2013-03-22 16:57:04

Last modified on

2013-03-22 16:57:04

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

13

Author

Wkbj79 (1863)

Entry type

Example

Classification

msc 11R29

Classification

msc 11R04

Classification

msc 13G05

Generated on Fri Feb 9 15:43:57 2018 by

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extensions without unramified subextensions and class number divisibility

extensions without unramified subextensions and class number divisibility

Theorem 1

.

Let

F

/

K

be an

extension

of

number fields

such that for any intermediate

Galois extension

L

/

K

, with

K

⊊

L

⊊

F

, there is at least one

finite place

or infinite place which ramifies in the extension

L

/

K

. Then,

h

K

, the

class number

of

K

, divides the class number of

F

,

h

F

.

First, we deduce some immediate

corollaries

.

Corollary 1

.

Let

F

/

K

be an extension of number fields which is

totally ramified

at some prime (or at an archimedean place). Then

h

K

divides

h

F

.

Proof.

The proof is clear since there cannot be

unramified

subextensions. The theorem applies.
∎

Corollary 2

.

Let

F

/

K

be a Galois extension of number fields such that

Gal

⁡

(

F

/

K

)

is a

non-abelian

simple group

. Then

h

K

divides

h

F

.

Proof.

In this case, there cannot be subextensions with

abelian

Galois group

and the theorem applies.
∎

Proof of the Theorem.

Let

H

be the

Hilbert class field

of

K

. By

definition

,

H

is the

maximal

unramified

abelian extension

of

K

,

Gal

⁡

(

H

/

K

)

is

isomorphic

to

Cl

⁡

(

K

)

, the ideal class group of

K

and

[

H

:

K

]

=

h

K

. Since there are no nontrivial unramified abelian subextensions of

F

/

K

, we have

F

∩

H

=

K

and so

[

F

H

:

F

]

=

[

H

:

K

]

=

h

K

. One can show that the extension

F

⁢

H

/

F

is unramified and abelian (in fact

Gal

⁡

(

F

⁢

H

/

F

)

≅

Gal

⁡

(

H

/

K

)

). Therefore

F

⁢

H

is contained in

L

, the Hilbert class field of

F

. Hence:

h

F

=

[

L

:

F

]

=

[

L

:

F

H

]

⋅

[

F

H

:

F

]

=

[

L

:

F

H

]

⋅

[

H

:

K

]

=

[

L

:

F

H

]

⋅

h

K

and so,

h

K

divides

h

F

.
∎

Title

extensions without unramified subextensions and class number divisibility

Canonical

name

ExtensionsWithoutUnramifiedSubextensionsAndClassNumberDivisibility

Date of creation

2013-03-22 15:07:35

Last modified on

2013-03-22 15:07:35

Owner

alozano (2414)

Last modified by

alozano (2414)

Numerical id

4

Author

alozano (2414)

Entry type

Theorem

Classification

msc 11R29

Classification

msc 11R32

Classification

msc 11R37

Related topic

PushDownTheoremOnClassNumbers

Related topic

ClassNumberDivisibilityInExtensions

Related topic

IdealClass

Related topic

ExistenceOfHilbertClassField

Related topic

CompositumOfAGaloisExtensionAndAnotherExtensionIsGalois

Related topic

DecompositionGroup

Generated on Fri Feb 9 15:44:00 2018 by

LaTeXML
Herbrand’s theorem

Herbrand’s theorem

Let

ℚ

⁢

(

ζ

p

)

be a

cyclotomic extension

of

ℚ

, with

p

an

odd prime

, let

A

be the Sylow

p

-

subgroup

of the

ideal class group

of

ℚ

⁢

(

ζ

p

)

, and let

G

be the

Galois group

of this

extension

. Note that the character group of

G

, denoted

G

^

, is given by

G

^

=

{

χ

i

∣

0

≤

i

≤

p

-

2

}

For each

χ

∈

G

^

, let

ε

χ

denote the corresponding

orthogonal idempotent

of the

group ring

, and note that the

p

-

Sylow subgroup

of the ideal class group is a

ℤ

⁢

[

G

]

-module under the typical

multiplication

. Thus, using the orthogonal idempotents, we can decompose the module

A

via

A

=

∑

i

=

0

p

-

2

A

ω

i

≡

∑

i

=

0

p

-

2

A

i

.

Last, let

B

k

denote the

k

th

Bernoulli number

.

Theorem 1

(Herbrand).

Let

i

be odd with

3

≤

i

≤

p

-

2

. Then

A

i

≠

0

⇔

p

∣

B

p

-

i

.

Only the first direction of this

theorem

(

⟹

) was proved by Herbrand himself. The converse is much more intricate, and was proved by Ken Ribet.

Title

Herbrand’s theorem

Canonical

name

HerbrandsTheorem

Date of creation

2013-03-22 14:12:45

Last modified on

2013-03-22 14:12:45

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

5

Author

mathcam (2727)

Entry type

Theorem

Classification

msc 11R29

Generated on Fri Feb 9 15:44:02 2018 by

LaTeXML
Hermite’s theorem

Hermite’s theorem

The following is a corollary of Minkowski’s theorem on

ideal classes

, which is a corollary of Minkowski’s theorem on lattices.

Definition

.

Let

S

=

{

p

1

,

…

,

p

r

}

be a set of

rational primes

p

i

∈

Z

. We say that a

number field

K

is

unramified

outside

S

if any prime not in

S

is unramified in

K

. In other words, if

p

is ramified in

K

, then

p

∈

S

. In other words, the only primes that divide the

discriminant

of

K

are elements of

S

.

Corollary

(Hermite’s Theorem).

Let

S

=

{

p

1

,

…

,

p

r

}

be a set of rational primes

p

i

∈

Z

and let

N

∈

N

be arbitrary. There is only a finite number of fields

K

which are unramified outside

S

and

bounded

degree

[

K

:

Q

]

≤

N

.

Title

Hermite’s theorem

Canonical name

HermitesTheorem

Date of creation

2013-03-22 15:05:35

Last modified on

2013-03-22 15:05:35

Owner

alozano (2414)

Last modified by

alozano (2414)

Numerical id

5

Author

alozano (2414)

Entry type

Corollary

Classification

msc 11R29

Classification

msc 11H06

Defines

unramified outside a set of primes

Generated on Fri Feb 9 15:44:03 2018 by

LaTeXML
ideal class

ideal class

Let

K

be a

number field

. Let

𝔞

and

𝔟

be ideals in

𝒪

K

(the

ring of algebraic integers

of

K

). Define a

relation

∼

on the ideals of

𝒪

K

in the following way: write

𝔞

∼

𝔟

if there exist nonzero

elements

α

and

β

of

𝒪

K

such that

(

α

)

⁢

𝔞

=

(

β

)

⁢

𝔟

.

The relation

∼

is an

equivalence relation

, and the

equivalence classes

under

∼

are known as

ideal classes

.

The number of equivalence classes, denoted by

h

or

h

K

, is called the

class number

of

K

.

Note that the set of ideals of any ring

R

forms an

abelian semigroup

with the

product of ideals

as the

semigroup

operation

. By replacing ideals by ideal classes, it is possible to define a group on the ideal classes of

𝒪

K

in the following way.

Let

𝔞

,

𝔟

be ideals of

𝒪

K

. Denote the ideal classes of which

𝔞

and

𝔟

are representatives by

[

𝔞

]

and

[

𝔟

]

respectively. Then define

⋅

by

[

𝔞

]

⋅

[

𝔟

]

=

[

𝔞

⁢

𝔟

]

Let

𝒞

=

{

[

𝔞

]

∣

𝔞

≠

(

0

)

,

𝔞

⁢

an ideal of

⁢

𝒪

K

}

.
With the above definition of

multiplication

,

𝒞

is an

abelian group

, called the

ideal class group

(or frequently just the

class group

) of

K

.

Note that the ideal class group of

K

is simply the

quotient group

of the ideal group of

K

by the

subgroup

of principal

fractional ideals

.

Title

ideal class

Canonical name

IdealClass

Date of creation

2013-03-22 12:36:42

Last modified on

2013-03-22 12:36:42

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

22

Author

mathcam (2727)

Entry type

Definition

Classification

msc 11R29

Synonym

ideal classes

Related topic

ExistenceOfHilbertClassField

Related topic

FractionalIdeal

Related topic

NumberField

Related topic

UnramifiedExtensionsAndClassNumberDivisibility

Related topic

ClassNumberDivisibilityInExtensions

Related topic

PushDownTheoremOnClassNumbers

Related topic

MinkowskisConstant

Related topic

ExtensionsWithoutUnramifiedSubextensionsAndClassNumberDivis

Defines

class number

Defines

ideal class group

Defines

class group

Generated on Fri Feb 9 15:44:06 2018 by

LaTeXML