aarjaneiro/planetmath huggingface dataset:
cyclotomic field

cyclotomic field

A

cyclotomic field

(or

cyclotomic number field

) is a

cyclotomic extension

of

ℚ

. These are all of the form

ℚ

⁢

(

ω

n

)

, where

ω

n

is a

primitive

n

th

root of unity

(

http://planetmath.org/PrimitiveNthRootOfUnity

).

The

ring of integers

of a cyclotomic field always has a

power basis

over

ℤ

(

http://planetmath.org/PowerBasisOverMathbbZ

). Specifically, the ring of integers of

ℚ

⁢

(

ω

n

)

is

ℤ

⁢

[

ω

n

]

.

Given a

ω

n

, its

minimal polynomial

over

ℚ

is the

cyclotomic polynomial

Φ

n

⁢

(

x

)

. Thus,

[

ℚ

(

ω

n

)

:

ℚ

]

=

φ

(

n

)

, where

φ

denotes the

Euler phi function

.

If

n

is odd, then

ℚ

⁢

(

ω

2

⁢

n

)

=

ℚ

⁢

(

ω

n

)

. There are many ways to prove this, but the following is a relatively short proof: Since

ω

n

=

ω

2

⁢

n

2

∈

ℚ

⁢

(

ω

2

⁢

n

)

, we have that

ℚ

⁢

(

ω

n

)

⊆

ℚ

⁢

(

ω

2

⁢

n

)

. We also have that

[

ℚ

(

ω

2

⁢

n

)

:

ℚ

]

=

φ

(

2

n

)

=

φ

(

2

)

φ

(

n

)

=

φ

(

n

)

=

[

ℚ

(

ω

n

)

:

ℚ

]

. Thus,

[

ℚ

(

ω

2

⁢

n

)

:

ℚ

(

ω

n

)

]

=

1

. It follows that

ℚ

⁢

(

ω

2

⁢

n

)

=

ℚ

⁢

(

ω

n

)

.

Note.

If

n

is a

positive integer

and

m

is an

integer

such that

gcd

⁡

(

m

,

n

)

=

1

, then

ω

n

and

ω

n

m

are the same cyclotomic field.

Title

cyclotomic field

Canonical

name

CyclotomicField

Date of creation

2013-03-22 17:10:25

Last modified on

2013-03-22 17:10:25

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

9

Author

Wkbj79 (1863)

Entry type

Definition

Classification

msc 11R18

Classification

msc 11-00

Synonym

cyclotomic number field

Related topic

CyclotomicExtension

Related topic

CyclotomicPolynomial

Generated on Fri Feb 9 15:43:09 2018 by

LaTeXML
cyclotomic units

cyclotomic units

Let

L

=

ℚ

⁢

(

ζ

m

)

be a

cyclotomic extension

of

ℚ

with

m

chosen to be minimal. Then the

ring of integers

is given by

ℤ

⁢

(

ζ

m

)

, and we denote the

group of units

by

𝒪

L

×

.

The

cyclotomic units

are the elements of a subgroup

C

of

𝒪

L

×

given by

η

=

ζ

m

r

-

1

ζ

m

s

-

1

with

r

and

s

relatively prime to

m

.

Title

cyclotomic units

Canonical

name

CyclotomicUnits

Date of creation

2013-03-22 14:12:36

Last modified on

2013-03-22 14:12:36

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

5

Author

mathcam (2727)

Entry type

Definition

Classification

msc 11R18

Generated on Fri Feb 9 15:43:11 2018 by

LaTeXML
examples of regular primes

examples of regular primes

Examples

:

1.

These are all the

irregular primes

up to

1061

:

37, 59, 67, 101, 103, 131, 149, 157, 233, 257, 263, 271,

283, 293, 307, 311, 347, 353, 379, 389, 401,

409, 421, 433, 461, 463, 467, 491, 523, 541,

547, 557, 577, 587, 593, 607, 613, 617, 619,

631, 647, 653, 659, 673, 677, 683, 691, 727,

751, 757, 761, 773, 797, 809, 811, 821, 827,

839, 877, 881, 887, 929, 953, 971, 1061.

(for this, see the

http://www.research.att.com/ njas/

sequences

/Seis.html

On-Line Encyclopedia of Integer Sequences

,

http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.c
gi?Anum=A000928

sequence A000928)

2.

The following are the first few

class numbers

of the

cyclotomic fields

ℚ

⁢

(

ζ

p

)

, where

ζ

p

is a primitive

p

-th

root of unity

:

p

Class Number

3

1

5

1

7

1

11

1

13

1

17

1

19

1

23

3

29

8

31

9

37

37

41

121

43

211

47

695

53

4889

59

41241

61

76301

An excellent reference for this is

[

1

]

.

Remarks

:

–

Notice that

37

divides

37

, and

59

divides

41241

=

3

⋅

59

⋅

233

, thus

37

,

59

are irregular primes (see above).

–

The class number of the cyclotomic fields grows very quickly with

p

. For example,

p

=

19

is the last cyclotomic field of class number 1.

References

1

L. C. Washington,

Introduction to Cyclotomic Fields

,
Springer-Verlag, New York.

Title

examples of regular primes

Canonical

name

ExamplesOfRegularPrimes

Date of creation

2013-03-22 14:05:58

Last modified on

2013-03-22 14:05:58

Owner

alozano (2414)

Last modified by

alozano (2414)

Numerical id

10

Author

alozano (2414)

Entry type

Example

Classification

msc 11R18

Classification

msc 11R29

Related topic

ClassNumbersAndDiscriminantsTopicsOnClassGroups

Generated on Fri Feb 9 15:43:13 2018 by

LaTeXML
prime ideal decomposition in cyclotomic extensions of ℚ

prime ideal decomposition in cyclotomic extensions of

ℚ

Let

q

∈

ℤ

be a prime greater than

2

, let

ζ

q

=

e

2

⁢

π

⁢

i

/

q

and write

L

=

ℚ

⁢

(

ζ

q

)

for the

cyclotomic extension

.
The

ring of integers

of

L

is

𝒪

L

=

ℤ

⁢

[

ζ

q

]

. The

discriminant

of

L

/

ℚ

is:

D

L

/

ℚ

=

±

q

q

-

2

and it is

+

exactly when

q

-

1

≡

0

,

1

⁢

mod

⁡

4

.

Proposition 1

.

±

q

∈

ℚ

⁢

(

ζ

q

)

, with

+

exactly when

q

-

1

≡

0

,

1

⁢

mod

⁡

4

.

Proof.

It can be proved that:

D

L

/

ℚ

=

±

q

q

-

2

=

∏

1

≤

i

<

j

≤

q

-

1

(

ζ

q

i

-

ζ

q

j

)

2

Taking

square roots

we obtain

q

q

-

3

2

⁢

±

q

=

∏

1

≤

i

<

j

≤

q

-

1

(

ζ

q

i

-

ζ

q

j

)

∈

ℚ

⁢

(

ζ

q

)

Hence the result
holds (and the sign depends on whether

q

-

1

≡

0

,

1

⁢

mod

⁡

4

).
∎

Let

K

=

ℚ

⁢

(

±

q

)

with the corresponding sign. Thus, by
the proposition we have a tower of fields:

\xymatrix

⁢

&

⁢

L

=

ℚ

⁢

(

ζ

q

)

⁢

\ar

⁢

@

-

[

d

]

⁢

&

⁢

K

⁢

\ar

⁢

@

-

[

d

]

⁢

&

⁢

ℚ

For a

prime ideal

p

⁢

ℤ

the decomposition in the

quadratic
extension

K

/

ℚ

is well-known (see

http://planetmath.org/encyclopedia/PrimeIdealDecompositionInQuadraticExtensionsOfMathbbQ.html

this entry). The next theorem
characterizes the decomposition in the

extension

L

/

ℚ

:

Theorem 1

.

Let

p

∈

Z

be a prime.

1.

If

p

=

q

,

q

⁢

𝒪

L

=

(

1

-

ζ

q

)

q

-

1

. In
other words, the prime

q

is

totally ramified

in

L

.

2.

If

p

≠

q

then

p

⁢

ℤ

splits into

(

q

-

1

)

/

f

distinct
primes in

𝒪

L

, where

f

is the order of

p

⁢

mod

⁡

q

(i.e.

p

f

≡

1

⁢

mod

⁡

q

,
and for all

1

<

n

<

f

,

p

n

≠

1

⁢

mod

⁡

q

).

References

1

Daniel A.Marcus,

Number Fields

. Springer, New York.

Title

prime ideal decomposition in cyclotomic extensions of

ℚ

Canonical name

PrimeIdealDecompositionInCyclotomicExtensionsOfmathbbQ

Date of creation

2013-03-22 13:53:49

Last modified on

2013-03-22 13:53:49

Owner

alozano (2414)

Last modified by

alozano (2414)

Numerical id

5

Author

alozano (2414)

Entry type

Theorem

Classification

msc 11R18

Related topic

PrimeIdealDecompositionInQuadraticExtensionsOfMathbbQ

Related topic

CalculatingTheSplittingOfPrimes

Related topic

KroneckerWeberTheorem

Related topic

ExamplesOfPrimeIdealDecompositionInNumberFields

Related topic

SplittingAndRamificationInNumberFieldsAndGaloisExtensions

Generated on Fri Feb 9 15:43:20 2018 by

LaTeXML
regular prime

regular prime

A prime

p

is

regular

if the

class number

of the

cyclotomic field

ℚ

⁢

(

ζ

p

)

is not

divisible

by

p

(where

ζ

p

:=

e

2

⁢

π

⁢

i

/

p

denotes a primitive

p

th

root of unity

). An

irregular prime

is a prime that is not regular.

Regular primes

rose to prominence as a result of Ernst Kummer’s work in the 1850’s on Fermat’s Last

Theorem

. Kummer was able to prove Fermat’s Last Theorem in the case where the exponent is a regular prime, a result that prior to Wiles’s recent work was the only demonstration of Fermat’s Last Theorem for a large class of exponents. In the course of this work Kummer also established the following numerical criterion for determining whether a prime is regular:

•

p

is regular if and only if none of the

numerators

of the

Bernoulli numbers

B

0

,

B

2

,

B

4

,

…

,

B

p

-

3

is a

multiple

of

p

.

Based on this criterion it is possible to give a heuristic

argument

that the regular primes have density

e

-

1

/

2

in the set of all primes

[

1

]

. Despite this, there is no known proof that the set of regular primes is

infinite

, although it is known that there are infinitely many irregular primes.

References

1

Kenneth Ireland & Michael Rosen,

A
Classical

Introduction

to Modern

Number Theory

,

Springer-Verlag, New
York, Second Edition, 1990.

Title

regular prime

Canonical

name

RegularPrime

Date of creation

2013-03-22 12:44:20

Last modified on

2013-03-22 12:44:20

Owner

djao (24)

Last modified by

djao (24)

Numerical id

6

Author

djao (24)

Entry type

Definition

Classification

msc 11R18

Classification

msc 11R29

Defines

irregular prime

Generated on Fri Feb 9 15:43:21 2018 by

LaTeXML
the cyclotomic units are algebraic units

the cyclotomic units are algebraic units

Let

L

=

ℚ

⁢

(

ζ

m

)

be a

cyclotomic extension

of

ℚ

with

m

chosen to be minimal and let

𝒪

L

be the

ring of integers

(

=

ℤ

⁢

(

ζ

m

)

), recall that

the

cyclotomic units

are the elements of the form

η

=

ζ

r

-

1

ζ

s

-

1

with

r

and

s

relatively prime to

m

(where

ζ

=

ζ

m

). Here we prove that these elements are indeed

algebraic units

, i.e.

η

∈

𝒪

L

×

.

Lemma 1

.

The

cyclotomic units are algebraic units

.

Proof.

In order to prove the lemma, we will check that both

η

and

η

-

1

are

algebraic integers

, thus

η

is a unit. Notice that it suffices to prove that

η

is an algebraic integer, because the rest follows from interchanging the role of

r

and

s

.

Let

r

,

s

∈

ℤ

be relatively prime to

m

, thus

r

mod

m

,

s

mod

m

are units in

ℤ

/

m

⁢

ℤ

and we can find an

integer

a

such that:

a

⋅

s

≡

r

mod

m

Note also that it follows that

ζ

r

=

ζ

a

⁢

s

. Moreover, using the equality of

polynomials

:

x

a

⁢

s

-

1

=

(

x

s

-

1

)

⋅

(

x

s

⁢

(

a

-

1

)

+

x

s

⁢

(

a

-

2

)

+

…

+

x

s

+

1

)

we get:

η

=

ζ

r

-

1

ζ

s

-

1

=

ζ

a

⁢

s

-

1

ζ

s

-

1

=

(

ζ

s

-

1

)

⋅

(

ζ

s

⁢

(

a

-

1

)

+

ζ

s

⁢

(

a

-

2

)

+

…

+

ζ

s

+

1

)

ζ

s

-

1

=

ζ

s

⁢

(

a

-

1

)

+

ζ

s

⁢

(

a

-

2

)

+

…

+

ζ

s

+

1

∈

𝒪

L

=

ℤ

⁢

[

ζ

]

Hence the result.
∎

Title

the cyclotomic units are algebraic units

Canonical name

TheCyclotomicUnitsAreAlgebraicUnits

Date of creation

2013-03-22 14:13:17

Last modified on

2013-03-22 14:13:17

Owner

alozano (2414)

Last modified by

alozano (2414)

Numerical id

4

Author

alozano (2414)

Entry type

Theorem

Classification

msc 11R18

Related topic

AlgebraicInteger

Generated on Fri Feb 9 15:43:24 2018 by

LaTeXML
Kronecker-Weber theorem

Kronecker-Weber theorem

The following theorem classifies the possible

http://planetmath.org/node/AbelianExtension

abelian extensions

of

ℚ

.

Theorem 1

(Kronecker-Weber Theorem).

Let

L

/

Q

be a finite

http://planetmath.org/node/AbelianExtension

abelian extension, then

L

is contained
in a

cyclotomic extension

, i.e. there is a

root of unity

ζ

such that

L

⊆

Q

⁢

(

ζ

)

.

In a similar fashion to this result, the theory of

elliptic curves

with

complex multiplication

provides a classification of

http://planetmath.org/node/AbelianExtension

abelian
extensions of quadratic imaginary number fields

:

Theorem 2

.

Let

K

be a

quadratic imaginary number field

with

ring of
integers

O

K

. Let

E

be an elliptic curve with
complex multiplication by

O

K

and let

j

⁢

(

E

)

be the

j

-invariant of

E

. Then:

1.

K

⁢

(

j

⁢

(

E

)

)

is the

Hilbert class field

of

K

.

2.

If

j

⁢

(

E

)

≠

0

,

1728

then the maximal

http://planetmath.org/node/AbelianExtension

abelian extension of

K

is given by:

K

a

⁢

b

=

K

⁢

(

j

⁢

(

E

)

,

h

⁢

(

E

torsion

)

)

where

h

⁢

(

E

torsion

)

is the set of

x

-coordinates of all the

torsion

points of

E

.

Note: The map

h

:

E

→

ℂ

is called a

Weber
function

for

E

. We can define a Weber function for the cases

j

⁢

(

E

)

=

0

,

1728

so the theorem holds true for those two cases as
well. Assume

E

:

y

2

=

x

3

+

A

⁢

x

+

B

, then:

h

⁢

(

P

)

=

{

x

⁢

(

P

)

,

if

j

⁢

(

E

)

≠

0

,

1728

;

x

2

⁢

(

P

)

,

if

j

⁢

(

E

)

=

1728

;

x

3

⁢

(

P

)

,

if

j

⁢

(

E

)

=

0

.

References

1

S. Lang,

Algebraic Number Theory

, Springer-Verlag, New York.

2

Joseph H. Silverman,

Advanced Topics in the

Arithmetic of Elliptic Curves

. Springer-Verlag, New
York.

Title

Kronecker-Weber theorem

Canonical name

KroneckerWeberTheorem

Date of creation

2013-03-22 13:52:41

Last modified on

2013-03-22 13:52:41

Owner

alozano (2414)

Last modified by

alozano (2414)

Numerical id

6

Author

alozano (2414)

Entry type

Theorem

Classification

msc 11R20

Classification

msc 11R37

Classification

msc 11R18

Related topic

ComplexMultiplication

Related topic

AbelianExtension

Related topic

PrimeIdealDecompositionInCyclotomicExtensionsOfMathbbQ

Related topic

NumberField

Related topic

CyclotomicExtension

Related topic

ArithmeticOfEllipticCurves

Defines

abelian extensions of quadratic imaginary number fields

Defines

Weber function

Generated on Fri Feb 9 15:43:26 2018 by

LaTeXML
fundamental units

fundamental units

The ring

R

of

algebraic integers

of any

algebraic number
field

contains a

finite set

H

=

{

η

1

,

η

2

,

…

,

η

t

}

of so-called

fundamental units

such that every unit

ε

of

R

is a power (

http://planetmath.org/GeneralAssociativity

)

product

of
these, multiplied by a

root of unity

:

ε

=

ζ

⋅

η

1

k

1

⁢

η

2

k

2

⁢

…

⁢

η

t

k

t

Conversely

, every such

element

ε

of the field is a
unit of

R

.

Examples:

units of quadratic fields

,  units of certain cubic fields (

http://planetmath.org/UnitsOfRealCubicFieldsWithExactlyOneRealEmbedding

)

For some algebraic number fields, such as all

imaginary
quadratic fields

, the set

H

may be empty (

t

=

0

).  In the
case of a single fundamental unit (

t

=

1

), which occurs e.g.
in all

real quadratic fields

(

http://planetmath.org/ImaginaryQuadraticField

),
there are two alternative units

η

and its

conjugate

η

¯

which one can use as
fundamental unit; then we can speak of

the

uniquely
determined fundamental unit

η

1

which is greater than 1.

Title

fundamental units

Canonical name

FundamentalUnits

Date of creation

2014-11-24 16:38:36

Last modified on

2014-11-24 16:38:36

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

22

Author

pahio (2872)

Entry type

Definition

Classification

msc 11R27

Classification

msc 11R04

Related topic

NumberField

Related topic

AlgebraicInteger

Generated on Fri Feb 9 15:43:28 2018 by

LaTeXML
Leopoldt’s conjecture

Leopoldt’s conjecture

Let

K

be a

number field

, and let

p

be a

rational prime

. Then

R

p

⁢

(

K

)

≠

0

, where

R

p

⁢

(

K

)

denotes the

p

-adic

regulator

(

http://planetmath.org/PAdicRegulator

) of

K

.

Though unproven for number fields in general, it is known to be true for

abelian extensions

of

ℚ

, and for certain

non-abelian

2-extensions of

imaginary quadratic extensions

of

ℚ

.

References

1

L. C. Washington,

Introduction to

Cyclotomic Fields

,
Springer-Verlag, New York.

Title

Leopoldt’s conjecture

Canonical name

LeopoldtsConjecture

Date of creation

2013-03-22 14:14:28

Last modified on

2013-03-22 14:14:28

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

6

Author

mathcam (2727)

Entry type

Conjecture

Classification

msc 11R27

Generated on Fri Feb 9 15:43:30 2018 by

LaTeXML
p-adic regulator

p

-adic regulator

Let

K

,

n

,

r

1

,

r

2

,

{

ε

n

,

…

,

ε

r

-

1

}

, and

|

|

⋅

|

|

i

be as in the entry

regulator

, but with

K

taken to be a CM field.

Define the

p

-adic logarithm

log

p

:

ℂ

p

×

→

ℂ

p

by

log

p

⁡

(

x

)

=

-

∑

k

=

1

∞

(

1

-

x

)

k

k

Let

A

K

,

p

be the

(

r

-

1

)

×

(

r

-

1

)

matrix with general entry given by

a

i

,

j

=

log

p

⁢

||

ε

j

||

i

. The

absolute value

of the

determinant

of this matrix is again independent of your choice of basis for the units and of the ordering of the

embeddings

. This value is called the

p

-adic regulator of

K

,

and is denoted by

R

p

,

K

, or

R

p

⁢

(

K

)

.

References

1

L. C. Washington,

Introduction

to

Cyclotomic Fields

,
Springer-Verlag, New York.

Title

p

-adic regulator

Canonical name

PadicRegulator

Date of creation

2013-03-22 14:14:13

Last modified on

2013-03-22 14:14:13

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

5

Author

mathcam (2727)

Entry type

Definition

Classification

msc 11R27

Related topic

PAdicExponentialAndPAdicLogarithm

Defines

p

-adic logarithm

Generated on Fri Feb 9 15:43:32 2018 by

LaTeXML
regulator

regulator

Let

K

be a

number field

with

[

K

:

ℚ

]

=

n

=

r

1

+

2

r

2

. Here

r

1

denotes the number of

real embeddings

:

σ

i

:

K

↪

ℝ

,

1

≤

i

≤

r

1

while

r

2

is half of the number of complex embeddings:

τ

j

:

K

↪

ℂ

,

1

≤

j

≤

r

2

Note that

{

τ

j

,

τ

¯

j

∣

1

≤

j

≤

r

2

}

are

all

the complex embeddings of

K

. Let

r

=

r

1

+

r

2

and for

1

≤

i

≤

r

define the “norm” in

K

corresponding to each
embedding:

∥

⋅

∥

i

:

K

×

→

ℝ

+

∥

α

∥

i

=

∣

σ

i

⁢

(

α

)

∣

,

1

≤

i

≤

r

1

∥

α

∥

r

1

+

j

=

∣

τ

j

⁢

(

α

)

∣

2

,

1

≤

j

≤

r

2

Let

𝒪

K

be the

ring of integers

of

K

. By Dirichlet’s unit theorem, we know that the rank of the

unit group

𝒪

K

×

is exactly

r

-

1

=

r

1

+

r

2

-

1

.
Let

{

ϵ

1

,

ϵ

2

,

…

,

ϵ

r

-

1

}

be a

fundamental system

of

generators

of

𝒪

K

×

modulo

roots of unity

(this is, modulo the torsion subgroup). Let

A

be the

r

×

(

r

-

1

)

matrix

A

=

(

log

⁢

∥

ϵ

1

∥

1

log

⁢

∥

ϵ

2

∥

1

…

log

⁢

∥

ϵ

r

-

1

∥

1

log

⁢

∥

ϵ

1

∥

2

log

⁢

∥

ϵ

2

∥

2

…

log

⁢

∥

ϵ

r

-

1

∥

2

⋮

⋮

⋱

⋮

log

⁢

∥

ϵ

1

∥

r

log

⁢

∥

ϵ

2

∥

r

…

log

⁢

∥

ϵ

r

-

1

∥

r

)

and let

A

i

be the

(

r

-

1

)

×

(

r

-

1

)

matrix obtained
by deleting the

i

-th row from

A

,

1

≤

i

≤

r

. It can be
checked that the

determinant

of

A

i

,

det

⁡

A

i

, is independent
up to sign of the choice of fundamental system of generators of

𝒪

K

×

and is also independent of the choice of

i

.

Definition

.

The

regulator

of

K

is defined to be

Reg

K

=

∣

det

⁡

A

1

∣

The regulator is one of the main ingredients in the analytic

class number formula

for number fields.

References

1

Daniel A. Marcus,

Number Fields

,
Springer, New York.

2

Serge Lang,

Algebraic Number Theory

. Springer-Verlag, New York.

Title

regulator

Canonical name

Regulator

Date of creation

2013-03-22 13:54:34

Last modified on

2013-03-22 13:54:34

Owner

alozano (2414)

Last modified by

alozano (2414)

Numerical id

8

Author

alozano (2414)

Entry type

Definition

Classification

msc 11R27

Related topic

NumberField

Related topic

DirichletsUnitTheorem

Related topic

ClassNumberFormula

Related topic

RegulatorOfAnEllipticCurve

Defines

regulator of a number field

Generated on Fri Feb 9 15:43:34 2018 by

LaTeXML
an exact sequence for ray class groups

an exact sequence for ray class groups

Let

K

be a

number field

, let

𝒪

K

be its

ring of integers

and let

𝔪

be a

modulus

in

K

, i.e.

𝔪

=

𝔪

0

⁢

𝔪

∞

where

𝔪

0

is an

integral ideal

in

𝒪

K

and

𝔪

∞

is a product of real

infinite places

(i.e. real

archimedean

primes). Let

Cl

⁡

(

K

)

be the

ideal class group

of

K

and let

Cl

⁡

(

K

,

𝔪

)

be the

ray class group

of

K

of

conductor

𝔪

. Also, define

(

𝒪

K

/

𝔪

)

×

=

(

𝒪

K

/

𝔪

0

)

×

×

(

ℤ

/

2

⁢

ℤ

)

|

𝔪

∞

|

where

|

𝔪

∞

|

denotes the number of real places in

𝔪

. Finally, let

U

=

𝒪

K

×

be the

unit group

of

K

.

Proposition

.

The elements above fit in the following exact sequence:

U

⟶

(

𝒪

K

/

𝔪

)

×

⟶

Cl

⁡

(

K

,

𝔪

)

⟶

Cl

⁡

(

K

)

⟶

1

.

Example 1

.

Let

K

=

ℚ

. Thus,

Cl

⁡

(

ℚ

)

is trivial and

U

=

{

±

1

}

≅

ℤ

/

2

⁢

ℤ

. Let

𝔪

=

p

⁢

∞

where

p

>

2

is any prime. Then:

(

ℤ

/

𝔪

)

×

=

(

ℤ

/

p

⁢

ℤ

)

×

×

(

ℤ

/

2

⁢

ℤ

)

.

The exact sequence now reads:

ℤ

/

2

⁢

ℤ

⟶

(

ℤ

/

p

⁢

ℤ

)

×

×

(

ℤ

/

2

⁢

ℤ

)

⟶

Cl

⁡

(

ℚ

,

p

⁢

∞

)

⟶

1

.

Therefore,

Cl

⁡

(

ℚ

,

p

⁢

∞

)

≅

(

ℤ

/

p

⁢

ℤ

)

×

. In fact, as we know, the

http://planetmath.org/node/RayClassField

ray class field of

ℚ

of conductor

𝔪

=

p

⁢

∞

is the

cyclotomic field

ℚ

⁢

(

ζ

p

)

where

ζ

p

is any primitive

p

th

root of unity

. Moreover

Gal

⁡

(

ℚ

⁢

(

ζ

p

)

/

ℚ

)

≅

Cl

⁡

(

ℚ

,

p

⁢

∞

)

≅

(

ℤ

/

p

⁢

ℤ

)

×

.

Finally notice that the ray class group of

ℚ

of conductor

𝔪

=

p

is simply

(

ℤ

/

p

⁢

ℤ

)

×

/

{

±

1

}

which corresponds to the ray class field

ℚ

⁢

(

ζ

p

)

+

=

ℚ

⁢

(

ζ

p

+

ζ

p

-

1

)

, the maximal real

subfield

of

ℚ

⁢

(

ζ

p

)

.

Title

an

exact sequence for ray class groups

Canonical name

AnExactSequenceForRayClassGroups

Date of creation

2013-03-22 15:42:46

Last modified on

2013-03-22 15:42:46

Owner

alozano (2414)

Last modified by

alozano (2414)

Numerical id

6

Author

alozano (2414)

Entry type

Result

Classification

msc 11R29

Related topic

Modulus

Related topic

RayClassField

Related topic

ClassNumbersAndDiscriminantsTopicsOnClassGroups

Generated on Fri Feb 9 15:43:40 2018 by

LaTeXML
class number divisibility in cyclic extensions

class number divisibility in cyclic extensions

In this entry, the

class number

of a

number field

L

is denoted by

h

L

.

Theorem 1

.

Let

F

/

K

be a cyclic

Galois extension

of degree

n

. Let

p

be a prime such that

n

is not

divisible

by

p

, and assume that

p

does not divide

h

E

, the class number of any intermediate field

K

⊆

E

⊊

F

. If

p

divides

h

F

then

p

f

also divides

h

F

, where

f

is the

multiplicative order

of

p

modulo

n

.

Recall that the multiplicative order of

p

modulo

n

is a number

f

such that

p

f

≡

1

mod

n

and

p

m

is not

congruent

to

1

modulo

n

for any

1

≤

m

<

f

.

Corollary 1

.

Let

F

/

K

be a Galois extension such that

[

F

:

K

]

=

q

is a prime distinct from the prime

p

. Assume that

p

does not divide

h

K

. If

p

divides

h

F

then

p

f

divides

h

F

, where

f

is the multiplicative order of

p

modulo

q

.

Note that a Galois extension

F

/

K

of prime degree has no non-trivial subextensions.

Title

class number divisibility in cyclic extensions

Canonical name

ClassNumberDivisibilityInCyclicExtensions

Date of creation

2013-03-22 15:07:41

Last modified on

2013-03-22 15:07:41

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

IdealClass

Related topic

ClassNumbersAndDiscriminantsTopicsOnClassGroups

Generated on Fri Feb 9 15:43:41 2018 by

LaTeXML