aarjaneiro/planetmath huggingface dataset:
prime factorizations of some small highly composite numbers

prime factorizations of some small highly composite numbers

All

highly composite numbers

have

prime factorizations

containing

consecutive

prime numbers

, starting with 2, while the exponents form a

sequence

in

descending order

(the number 1 as an exponent is tacit in the following table).

HCN

Factorization

4

2

2

6

2

×

3

12

2

2

×

3

24

2

3

×

3

36

2

2

×

3

2

48

2

4

×

3

60

2

2

×

3

×

5

120

2

3

×

3

×

5

180

2

2

×

3

2

×

5

240

2

4

×

3

×

5

360

2

3

×

3

2

×

5

720

2

4

×

3

2

×

5

840

2

3

×

3

×

5

×

7

1260

2

2

×

3

2

×

5

×

7

1680

2

4

×

3

×

5

×

7

2520

2

3

×

3

2

×

5

×

7

5040

2

4

×

3

2

×

5

×

7

7560

2

3

×

3

3

×

5

×

7

10080

2

5

×

3

2

×

5

×

7

15120

2

4

×

3

3

×

5

×

7

20160

2

6

×

3

2

×

5

×

7

25200

2

4

×

3

2

×

5

2

×

7

27720

2

3

×

3

2

×

5

×

7

×

11

45360

2

4

×

3

4

×

5

×

7

50400

2

5

×

3

2

×

5

2

×

7

55440

2

4

×

3

2

×

5

×

7

×

11

83160

2

3

×

3

3

×

5

×

7

×

11

110880

2

5

×

3

2

×

5

×

7

×

11

166320

2

4

×

3

3

×

5

×

7

×

11

221760

2

6

×

3

2

×

5

×

7

×

11

277200

2

4

×

3

2

×

5

2

×

7

×

11

332640

2

5

×

3

3

×

5

×

7

×

11

498960

2

4

×

3

4

×

5

×

7

×

11

554400

2

5

×

3

2

×

5

2

×

7

×

11

665280

2

6

×

3

3

×

5

×

7

×

11

720720

2

4

×

3

2

×

5

×

7

×

11

×

13

1081080

2

3

×

3

3

×

5

×

7

×

11

×

13

1441440

2

5

×

3

2

×

5

×

7

×

11

×

13

2162160

2

4

×

3

3

×

5

×

7

×

11

×

13

Title

prime factorizations of some small highly composite numbers

Canonical name

PrimeFactorizationsOfSomeSmallHighlyCompositeNumbers

Date of creation

2013-03-22 18:51:29

Last modified on

2013-03-22 18:51:29

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

5

Author

PrimeFan (13766)

Entry type

Example

Classification

msc 11N56

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

LaTeXML
Chinese remainder theorem

Chinese remainder theorem

Let

R

be a

commutative ring

with

identity

. If

I

1

,

…

,

I

n

are ideals of

R

such that

I

i

+

I

j

=

R

whenever

i

≠

j

, then let

I

=

∩

i

=

1

n

I

i

=

∏

i

=

1

n

I

i

.

The sum of quotient maps

R

/

I

→

R

/

I

i

gives an

isomorphism

R

/

I

≅

∏

i

=

1

n

R

/

I

i

.

This has the slightly weaker

consequence

that given a system of

congruences

x

≅

a

i

(

mod

I

i

)

, there is a

solution

in

R

which is unique mod

I

, as the

theorem

is usually stated for the

integers

.

Title

Chinese remainder theorem

Canonical name

ChineseRemainderTheorem1

Date of creation

2013-03-22 12:16:43

Last modified on

2013-03-22 12:16:43

Owner

bwebste (988)

Last modified by

bwebste (988)

Numerical id

7

Author

bwebste (988)

Entry type

Theorem

Classification

msc 11N99

Classification

msc 11A05

Classification

msc 13A15

Related topic

ChineseRemainderTheoremInTermsOfDivisorTheory

Generated on Fri Feb 9 15:40:17 2018 by

LaTeXML
Lagrange’s four-square theorem

Lagrange’s four-square theorem

Lagrange’s four-square theorem states that every non-negative

integer

may be expressed as the sum of at most four squares. By the

Euler four-square identity

, it is enough to show that every prime is expressible by at most four squares. It was later proved that only the numbers of the form

4

n

⁢

(

8

⁢

m

+

7

)

require four squares.

This shows that

g

⁢

(

2

)

=

G

⁢

(

2

)

=

4

, where

g

and

G

are the

Waring functions

(

http://planetmath.org/WaringsProblem

).

Title

Lagrange’s four-square theorem

Canonical name

LagrangesFoursquareTheorem

Date of creation

2013-03-22 12:35:17

Last modified on

2013-03-22 12:35:17

Owner

bbukh (348)

Last modified by

bbukh (348)

Numerical id

10

Author

bbukh (348)

Entry type

Theorem

Classification

msc 11P05

Related topic

WaringsProblem

Related topic

EulerFourSquareIdentity

Generated on Fri Feb 9 15:40:19 2018 by

LaTeXML
Waring’s problem

Waring’s problem

Waring asked whether it is possible to represent every

natural number

as a sum of

bounded

(

http://

planetmath

.org/BoundedInterval

) number of nonnegative

k

’th powers, that is, whether the set

{

n

k

∣

n

∈

ℤ

+

}

is an

additive basis

(

http://planetmath.org/Basis2

). He was led to this conjecture by Lagrange’s

theorem

(

http://planetmath.org/LagrangesFourSquareTheorem

) which asserted that every natural number can be represented as a sum of four squares.

Hilbert

[

1

]

was the first to prove the conjecture for all

k

. In his paper he did not give an explicit bound on

g

⁢

(

k

)

, the number of powers needed, but later it was proved that

g

⁢

(

k

)

=

2

k

+

⌊

(

3

2

)

k

⌋

-

2

except possibly finitely many exceptional

k

, none of which are known.

Wooley

[

4

]

, improving the result of Vinogradov

[

3

]

, proved that the number of

k

’th powers needed to represent all

sufficiently large

integers

is

G

⁢

(

k

)

≤

k

⁢

(

ln

⁡

k

+

ln

⁡

ln

⁡

k

+

O

⁢

(

1

)

)

.

References

1

David Hilbert

.

Beweis für Darstellbarkeit der ganzen Zahlen durch eine feste
Anzahl

n

-ter Potenzen (Waringsches Problem).

Math. Ann.

, pages 281–300, 1909.

Available electronically from

http://gdz.sub.uni-goettingen.de/en/index.html

GDZ.

2

Robert C. Vaughan.

The Hardy-Littlewood method

.

Cambridge University Press, 1981.

http://www.emis.de/cgi-bin/zmen/ZMATH/en/quick.html?type=html&an=0868.11046

Zbl 0868.11046.

3

I. M. Vinogradov.

On an upper bound for

G

⁢

(

n

)

.

Izv. Akad. Nauk SSSR. Ser. Mat.

, 23:637–642, 1959.

http://www.emis.de/cgi-bin/zmen/ZMATH/en/quick.html?type=html&an=0089.02703

Zbl 0089.02703.

4

Trevor D. Wooley.

Large improvements in Waring’s problem.

Ann. Math

, 135(1):131–164, 1992.

http://www.emis.de/cgi-bin/zmen/ZMATH/en/quick.html?type=html&an=0754.11026

Zbl 0754.11026.

http://links.jstor.org/sici?sici=0003-486X%28199201%292%3A135%3A1%3C131%3ALIIWP%3E2.0.CO%3B2-O

Available online at

http://www.jstor.org

JSTOR.

Title

Waring’s problem

Canonical name

WaringsProblem

Date of creation

2013-03-22 13:19:46

Last modified on

2013-03-22 13:19:46

Owner

bbukh (348)

Last modified by

bbukh (348)

Numerical id

13

Author

bbukh (348)

Entry type

Theorem

Classification

msc 11P05

Classification

msc 11B13

Related topic

LagrangesFourSquareTheorem

Related topic

Basis2

Generated on Fri Feb 9 15:40:26 2018 by

LaTeXML
Goldbach’s conjecture

Goldbach’s conjecture

The conjecture states that every

even integer

n

>

2

is expressible as the sum of two primes.

In 1966 Chen proved that every sufficiently large

even number

can be expressed as the sum of a prime and a number with at most two

prime divisors

.

Vinogradov proved that every sufficiently large

odd

number is a sum of three primes. In 1997 it was shown by J.-M. Deshouillers, G. Effinger, H. Te Riele, and D. Zinoviev that, assuming a

generalized Riemann hypothesis

, every

odd number

n

>

5

can be represented as sum of three primes.

The conjecture was first proposed in a 1742 letter from Christian Goldbach to Euler and still remains unproved.

Title

Goldbach’s conjecture

Canonical name

GoldbachsConjecture

Date of creation

2013-03-22 12:13:43

Last modified on

2013-03-22 12:13:43

Owner

drini (3)

Last modified by

drini (3)

Numerical id

11

Author

drini (3)

Entry type

Conjecture

Classification

msc 11P32

Classification

msc 11-00

Related topic

Prime

Generated on Fri Feb 9 15:40:28 2018 by

LaTeXML
Levy’s conjecture

Levy’s conjecture

Conjecture

(Émile Lemoine). All

odd integers

greater than 5 can be represented as the sum of an

odd prime

and an even

semiprime

. In other words,

2

⁢

n

+

1

=

p

+

2

⁢

q

always has a solution in primes

p

and

q

(not necessarily distinct) for

n

>

2

.

For example,

47

=

13

+

2

×

17

=

37

+

2

×

5

=

41

+

2

×

3

=

43

+

2

×

2

. A046927 in Sloane’s OEIS counts how many different ways

2

⁢

n

+

1

can be represented as

p

+

2

⁢

q

.

The conjecture was first stated by Émile Lemoine in 1894. In 1963,

Hyman Levy

published a paper mentioning this conjecture in

relation

to Goldbach’s conjecture.

References

1

L. E. Dickson,

History of the Theory of Numbers

Vol. I. Providence, Rhode Island:

American Mathematical Society

& Chelsea Publications (1999): 424

2

R. K. Guy,

Unsolved Problems in

Number Theory

New York: Springer-Verlag 2004: C1

3

L. Hodges, “A lesser-known

Goldbach conjecture

”,

Math. Mag.

,

66

(1993): 45 - 47.

4

É. Lemoine, “title”

L’intermediaire des mathematiques

179

3 (1896): 151

5

H. Levy, “On Goldbach’s Conjecture”,

Math. Gaz.

47

(1963): 274

Title

Levy’s conjecture

Canonical

name

LevysConjecture

Date of creation

2013-03-22 17:26:32

Last modified on

2013-03-22 17:26:32

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

6

Author

PrimeFan (13766)

Entry type

Conjecture

Classification

msc 11P32

Synonym

Levy conjecture

Synonym

Lemoine’s conjecture

Generated on Fri Feb 9 15:40:30 2018 by

LaTeXML
part of a partition

part of a partition

If

λ

=

(

λ

1

,

λ

2

,

…

,

λ

k

)

is an

integer partition

, then each

λ

j

is a

part

of

λ

. The

length

of

λ

is defined as the number of its parts.
If

m

j

is the number of parts equal to

j

, then the

partition

λ

is also written as

λ

=

(

1

m

1

,

2

m

2

,

3

m

3

,

…

)

.

For example, if

λ

=

(

5

,

4

,

4

,

4

,

3

,

3

,

3

,

3

,

3

,

1

,

1

)

then we also write

λ

=

(

1

2

,

3

5

,

4

3

,

5

1

)

.

Title

part of a partition

Canonical

name

PartOfAPartition

Date of creation

2013-03-22 15:01:34

Last modified on

2013-03-22 15:01:34

Owner

drini (3)

Last modified by

drini (3)

Numerical id

5

Author

drini (3)

Entry type

Definition

Classification

msc 11P99

Classification

msc 05A17

Related topic

IntegerPartition

Defines

length

Generated on Fri Feb 9 15:40:35 2018 by

LaTeXML
algebraic conjugates

algebraic conjugates

Let

L

be an

algebraic extension

of a field

K

, and let

α

1

∈

L

be algebraic over

K

. Then

α

1

is the root of a

minimal polynomial

f

⁢

(

x

)

∈

K

⁢

[

x

]

. Denote the other roots of

f

⁢

(

x

)

in

L

by

α

2

,

α

3

,

…

,

α

n

. These (along with

α

1

itself) are the

algebraic conjugates

of

α

1

and any two are said to be

algebraically conjugate

.

The notion of

algebraic

conjugacy is a special case of group conjugacy in the case where the group in question is the

Galois group

of the above minimal polynomial, viewed as acting on the roots of said

polynomial

.

Title

algebraic conjugates

Canonical name

AlgebraicConjugates

Date of creation

2013-03-22 13:58:39

Last modified on

2013-03-22 13:58:39

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

9

Author

mathcam (2727)

Entry type

Definition

Classification

msc 11R04

Synonym

algebraically conjugate

Synonym

conjugate

Related topic

ComplexConjugate

Related topic

ConjugationMnemonic

Generated on Fri Feb 9 15:40:37 2018 by

LaTeXML
algebraic integer

algebraic integer

Let

K

be an

extension

(

http://planetmath.org/ExtensionField

) of

ℚ

contained

in

ℂ

. A number

α

∈

K

is called an

algebraic integer

of

K

if it is the root of a

monic polynomial

with

coefficients

in

ℤ

, i.e., an element of

K

that is integral over

ℤ

. Every algebraic integer is an

algebraic number

(with

K

=

ℂ

), but the

converse

is false.

Title

algebraic integer

Canonical

name

AlgebraicInteger

Date of creation

2013-03-22 11:45:41

Last modified on

2013-03-22 11:45:41

Owner

KimJ (5)

Last modified by

KimJ (5)

Numerical id

13

Author

KimJ (5)

Entry type

Definition

Classification

msc 11R04

Classification

msc 62-01

Classification

msc 03-01

Related topic

IntegralBasis

Related topic

CyclotomicUnitsAreAlgebraicUnits

Related topic

FundamentalUnits

Related topic

Monic2

Related topic

RingWithoutIrreducibles

Generated on Fri Feb 9 15:40:39 2018 by

LaTeXML
algebraic number

algebraic number

A number

α

∈

ℂ

is called an

algebraic number

if there exists a

polynomial

f

⁢

(

x

)

=

a

n

⁢

x

n

+

⋯

+

a

0

such that

a

0

,

…

,

a

n

, not all zero, are in

ℚ

and

f

⁢

(

α

)

=

0

.

Title

algebraic number

Canonical name

AlgebraicNumber

Date of creation

2013-03-22 11:45:38

Last modified on

2013-03-22 11:45:38

Owner

KimJ (5)

Last modified by

KimJ (5)

Numerical id

10

Author

KimJ (5)

Entry type

Definition

Classification

msc 11R04

Classification

msc 41A58

Classification

msc 41A50

Classification

msc 42B05

Classification

msc 42A16

Classification

msc 42C15

Classification

msc 18-00

Related topic

Pi

Related topic

Irrational

Related topic

AlgebraicElement

Related topic

DegreeOfAnAlgebraicNumber

Generated on Fri Feb 9 15:40:42 2018 by

LaTeXML
algebraic numbers are countable

algebraic numbers are countable

Theorem

.

The set of (a) all

algebraic numbers

, (b) the real algebraic numbers is

countable

.

Proof.

Let’s consider the

algebraic equations

P

⁢

(

x

)

=

0

(1)

where

P

⁢

(

x

)

:=

a

0

⁢

x

n

+

a

1

⁢

x

n

-

1

+

…

+

a

n

-

1

⁢

x

+

a

n

is an

irreducible

(

http://planetmath.org/IrreduciblePolynomial2

) and

primitive polynomial

with

integer

coefficients

a

j

and

a

0

>

0

.  Each algebraic number exactly one such equation (see the

minimal polynomial

).  For every integer

N

=

2

,

3

,

4

,

…

there exists a finite number of equations (1) such that

n

+

a

0

+

|

a

1

|

+

…

+

|

a

n

|

=

N

(e.g. if

N

=

3

,  then one has the equations

x

-

1

=

0

and

x

+

1

=

0

) and thus only a

finite set

of algebraic numbers as the of these equations.  These algebraic numbers may be ordered to a

finite sequence

(

http://planetmath.org/OrderedTuplet

)

S

N

using a system, for example by the magnitude of the

real part

and the

imaginary part

.  When one forms the concatenated

sequence

S

2

,

S

3

,

S

4

,

…

it comprises all algebraic numbers in a countable setting, which defines a

bijection

from the set onto

ℤ

+

.

References

1

E. Kamke:

Mengenlehre

.  Sammlung Göschen: Band 999/999a.  – Walter de Gruyter & Co., Berlin (1962).

Title

algebraic numbers are countable

Canonical name

AlgebraicNumbersAreCountable

Date of creation

2013-03-22 15:13:47

Last modified on

2013-03-22 15:13:47

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

14

Author

pahio (2872)

Entry type

Theorem

Classification

msc 11R04

Classification

msc 03E10

Synonym

algebraic numbers may be set in a sequence

Related topic

HeightOfAnAlgebraicNumber2

Related topic

ProofOfTheExistenceOfTranscendentalNumbers

Related topic

A_nAreCountableSoIsA_1XXA_nIfA_1

Related topic

ExamplesOfCountableSets

Related topic

FieldOfAlgebraicNumbers

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

LaTeXML
algebraic sines and cosines

algebraic sines and cosines

For any

rational number

r

, the sine and the cosine of the number

r

⁢

π

are

algebraic numbers

.

Proof.

According to the

http://

planetmath

.org/node/11664

parent entry,

sin

⁡

n

⁢

φ

and

cos

⁡

n

⁢

φ

can be expressed as

polynomials

with

integer

coefficients of

sin

⁡

φ

or

cos

⁡

φ

, respectively, when

n

is an integer.  Thus we can write

sin

⁡

n

⁢

φ

=

P

⁢

(

sin

⁡

φ

)

,

cos

⁡

n

⁢

φ

=

Q

⁢

(

cos

⁡

φ

)

,

where

P

⁢

(

x

)

,

Q

⁢

(

x

)

∈

ℤ

⁢

[

x

]

.  If

r

=

m

n

where

m

,

n

are integers and

n

≠

0

,  we have

P

⁢

(

sin

⁡

r

⁢

π

)

=

sin

⁡

n

⁢

r

⁢

π

=

sin

⁡

m

⁢

π

=

0

,

Q

⁢

(

cos

⁡

r

⁢

π

)

=

cos

⁡

n

⁢

r

⁢

π

=

cos

⁡

m

⁢

π

=

±

1

,

i.e. both

sin

⁡

r

⁢

π

and

cos

⁡

r

⁢

π

satisfy

an

algebraic equation

.  Q.E.D.

For example,

cos

⁡

7

⁢

φ

=

64

⁢

cos

7

⁡

φ

-

112

⁢

cos

5

⁡

φ

+

56

⁢

cos

3

⁡

φ

-

7

⁢

cos

⁡

φ

,

whence we have the

identity

64

⁢

cos

7

⁡

π

7

-

112

⁢

cos

5

⁡

π

7

+

56

⁢

cos

3

⁡

π

7

-

7

⁢

cos

⁡

π

7

+

1

=

0

,

and therefore

cos

⁡

π

7

is algebraic over

ℤ

.

Title

algebraic sines and cosines

Canonical name

AlgebraicSinesAndCosines

Date of creation

2013-03-22 18:51:27

Last modified on

2013-03-22 18:51:27

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

7

Author

pahio (2872)

Entry type

Corollary

Classification

msc 11R04

Classification

msc 11C08

Related topic

RationalSineAndCosine

Related topic

MultiplesOfAnAlgebraicNumber

Generated on Fri Feb 9 15:40:45 2018 by

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all algebraic numbers in a sequence

all algebraic numbers in a sequence

The beginning of the

sequence

of all

algebraic numbers

ordered as explained in the parent (

http://planetmath.org/AlgebraicNumbersAreCountable

) entry is as follows:

0

;

-

1

,

1

;

-

2

,

-

1

2

,

-

i

,

i

,

1

2

,

2

;

-

3

,

-

1

-

5

2

,

-

2

,

-

1

2

,

1

-

5

2

,

-

1

-

i

⁢

3

2

,

-

1

+

i

⁢

3

2

,

-

1

3

,

-

i

⁢

2

,

-

i

2

,

i

2

,

i

⁢

2

,

1

3

,

1

-

i

⁢

3

2

,

1

+

i

⁢

3

2

,

-

1

+

5

2

,

1

2

,

2

,

1

+

5

2

,

3

;

…

The first number corresponds to the

algebraic equation

x

=

0

,  the two following numbers to the equations

x

±

1

=

0

,  the six following to the equations

x

±

2

=

0

,

2

⁢

x

±

1

=

0

,

x

2

+

1

=

0

,  the twenty following to the equations

x

±

3

=

0

,

3

⁢

x

±

1

=

0

,

x

2

±

x

±

1

=

0

,

x

2

±

2

=

0

,

2

⁢

x

2

±

1

=

0

.

In practice, one cannot continue the sequence very far since the higher degree equations – quintic and so on – are non-solvable by

radicals

(

http://planetmath.org/NthRoot

); instead we can list the equations satisfied by the numbers as far we want and tell how many roots (

http://planetmath.org/Root

) they have.  In principle, the number sequence does exist!

Title

all algebraic numbers in a sequence

Canonical name

AllAlgebraicNumbersInASequence

Date of creation

2013-03-22 15:13:58

Last modified on

2013-03-22 15:13:58

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

11

Author

pahio (2872)

Entry type

Result

Classification

msc 11R04

Classification

msc 03E10

Synonym

counting the algebraic numbers

Related topic

FieldOfAlgebraicNumbers

Generated on Fri Feb 9 15:40:47 2018 by

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canonical basis

canonical basis

Let

ϑ

be an

algebraic integer

of degree (

http://planetmath.org/ExtensionField

)

n

.  The

algebraic number field

ℚ

⁢

(

ϑ

)

has always an

integral basis

of the form

ω

1

=

1

,

ω

2

=

a

21

+

ϑ

d

2

,

ω

3

=

a

31

+

a

32

⁢

ϑ

+

ϑ

2

d

3

,

⋮

⋮

⋮

ω

n

=

a

n

⁢

1

+

a

n

⁢

2

⁢

ϑ

+

…

+

a

n

,

n

-

1

⁢

ϑ

n

-

2

+

ϑ

n

-

1

d

n

,

where the

a

i

⁢

j

’s and

d

i

’s are

rational integers

such that

d

2

⁢

∣

d

3

∣

⁢

d

4

⁢

∣

…

∣

⁢

d

n

,

i.e.

d

i

∣

d

i

+

1

∀

i

=

2

,

3

,

…

,

n

-

1

.

The integral basis

ω

1

,

ω

2

,

…

,

ω

n

is called a

canonical basis

of the

number field

.

Remark.

The

integers

a

i

⁢

j

can be reduced so that for all

i

and

j

,

-

d

i

2

<

a

i

⁢

j

≦

d

i

2

.

Then one may speak of an

adjusted canonical basis

.  In the case of a quadratic number field

ℚ

⁢

(

d

)

with

d

≡

1

⁢

(

mod

⁢

4

)

we have (see the

examples of ring

of integers of a number field)

ω

1

=

1

,

ω

2

=

1

+

d

2

.

The

discriminant

of this basis is

d

.

Title

canonical basis

Canonical name

CanonicalBasis

Date of creation

2015-02-06 13:12:19

Last modified on

2015-02-06 13:12:19

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

14

Author

pahio (2872)

Entry type

Theorem

Classification

msc 11R04

Related topic

MinimalityOfIntegralBasis

Related topic

ExamplesOfRingOfIntegersOfANumberField

Related topic

ConditionForPowerBasis

Related topic

IntegralBasisOfQuadraticField

Related topic

CanonicalFormOfElementOfNumberField

Defines

canonical basis

Defines

canonical basis of a number field

Defines

adjusted canonical basis

Generated on Fri Feb 9 15:40:52 2018 by

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characterizing CM-fields using Dirichlet’s unit theorem

characterizing CM-fields using Dirichlet’s unit theorem

If

K

is a

number field

,

𝒪

K

is the

ring of algebraic integers

in

K

, and

𝒪

K

⋆

is the (

multiplicative

) group of units in

𝒪

K

. Dirichlet’s unit

theorem

gives the

structure

of the unit group. We can use that theorem to characterize CM-fields:

Theorem 1

.

Let

Q

⊂

F

⊂

K

be nontrivial

extensions

of number fields. Then

K

is a CM-field, with

F

its totally real

subfield

, if and only if

O

K

⋆

/

O

F

⋆

is finite.

We use the notation of the article on Dirichlet’s unit theorem, where

r

(and

r

F

,

r

K

) is used to count

real embeddings

and

s

(as well as

s

F

,

s

K

) to count complex embeddings, and we write

μ

⁢

(

F

)

or

μ

⁢

(

K

)

for the group of

roots of unity

in

𝒪

F

⋆

or

𝒪

K

⋆

.

Proof.

Write

n

=

[

F

:

ℚ

]

,

m

=

[

K

:

F

]

>

1

.

(

⇒

): If

K

/

F

is CM, then since

F

is totally real,

r

F

=

n

,

s

F

=

0

. Hence by Dirichlet’s unit theorem,

𝒪

F

⋆

≅

μ

⁢

(

F

)

×

ℤ

n

-

1

. Since

K

/

F

is a

complex

quadratic extension

,

[

K

:

ℚ

]

=

2

n

and all its

embeddings

are complex. Thus

r

K

=

0

,

2

⁢

s

K

=

2

⁢

n

. Hence

𝒪

K

⋆

≅

μ

⁢

(

K

)

×

ℤ

n

-

1

as well. Clearly

𝒪

F

⋆

⊂

𝒪

K

⋆

, and since they have the same rank (

http://planetmath.org/FreeModule

), their

quotient

is torsion and thus finite.

(

⇐

): Since

𝒪

K

⋆

/

𝒪

F

⋆

is finite, the ranks of these groups are equal and thus

r

F

+

s

F

=

r

K

+

s

K

again by Dirichlet’s unit theorem.

Now,

r

K

+

2

⁢

s

K

=

m

⁢

n

=

m

⁢

(

r

F

+

2

⁢

s

F

)

(1)

r

K

+

s

K

=

r

F

+

s

F

;

(2)

subtracting (2) from (1), we get

s

K

=

(

m

-

1

)

⁢

(

r

F

+

2

⁢

s

F

)

+

s

F

≥

(

m

-

1

)

⁢

n

(3)

and thus

m

⁢

n

=

r

K

+

2

⁢

s

K

≥

r

K

+

2

⁢

(

m

-

1

)

⁢

n

so that

0

≤

r

K

≤

n

⁢

(

2

-

m

)

. Thus

m

≤

2

, and since

K

is a nontrivial extension, we must have

m

=

2

so that

K

/

F

is quadratic and

r

K

=

0

(since

n

⁢

(

2

-

m

)

=

0

).

Finally, by (3), we then have

s

K

=

r

F

+

3

⁢

s

F

; (2) says that

s

K

=

r

F

+

s

F

, and thus

s

F

=

0

. It follows that

F

is totally real and, since

r

K

=

0

,

K

must be an

imaginary quadratic extension

of

F

.

Title

characterizing CM-fields using Dirichlet’s unit theorem

Canonical name

CharacterizingCMfieldsUsingDirichletsUnitTheorem

Date of creation

2013-03-22 17:57:26

Last modified on

2013-03-22 17:57:26

Owner

rm50 (10146)

Last modified by

rm50 (10146)

Numerical id

4

Author

rm50 (10146)

Entry type

Theorem

Classification

msc 11R04

Classification

msc 11R27

Classification

msc 12D99

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