aarjaneiro/planetmath huggingface dataset:
unramified extensions and class number divisibility

unramified extensions and class number divisibility

The following is a corollary of the

existence

of the

Hilbert class field

.

Corollary 1

.

Let

K

be a

number field

,

h

K

is its

class number

and let

p

be a prime. Then

K

has an everywhere

unramified

Galois extension

of degree

p

if and only if

h

K

is

divisible

by

p

.

Proof.

Let

K

be a number field and let

H

be the Hilbert class field of

K

. Then:

|

Gal

(

H

/

K

)

|

=

[

H

:

K

]

=

h

K

.

Let

p

be a

prime number

. Suppose that there exists a Galois extension

F

/

K

, such that

[

F

:

K

]

=

p

and

F

/

K

is everywhere unramified. Notice that any Galois extension of prime degree is

abelian

(because any group of prime degree

p

is abelian,

isomorphic

to

ℤ

/

p

⁢

ℤ

). Since

H

is the

maximal

abelian unramified

extension

of

K

the following

inclusions

occur:

K

⊊

F

⊆

H

Moreover,

h

K

=

[

H

:

K

]

=

[

H

:

F

]

⋅

[

F

:

K

]

=

[

H

:

F

]

⋅

p

.

Therefore

p

divides

h

K

.

Next we prove the remaining direction. Suppose that

p

divides

h

K

=

|

Gal

⁡

(

H

/

K

)

|

. Since

G

=

Gal

⁡

(

H

/

K

)

is an

abelian group

(isomorphic to the

class group

of

K

) there exists a

normal subgroup

J

of

G

such that

|

G

/

J

|

=

p

. Let

F

=

H

J

be the

fixed field

by the

subgroup

J

, which is, by the main

theorem

of

Galois theory

, a Galois extension of

K

. This field

satisfies

[

F

:

K

]

=

p

and, since

F

is included in

H

, the extension

F

/

K

is abelian and everywhere unramified, as claimed.
∎

Title

unramified extensions and class number divisibility

Canonical name

UnramifiedExtensionsAndClassNumberDivisibility

Date of creation

2013-03-22 15:02:59

Last modified on

2013-03-22 15:02:59

Owner

alozano (2414)

Last modified by

alozano (2414)

Numerical id

5

Author

alozano (2414)

Entry type

Corollary

Classification

msc 11R37

Classification

msc 11R32

Classification

msc 11R29

Related topic

IdealClass

Related topic

PExtension

Related topic

Ramify

Related topic

ClassNumbersAndDiscriminantsTopicsOnClassGroups

Generated on Fri Feb 9 15:45:23 2018 by

LaTeXML
values of Dedekind zeta functions of real quadratic number fields at negative integers

values of Dedekind zeta functions of real quadratic number fields at negative integers

Let

K

be a real

quadratic number field

of

discriminant

D

K

and let

ζ

⁢

(

s

,

K

)

be the

Dedekind zeta function

associated to

K

. By the

Siegel-Klingen Theorem

, if

n

>

0

then

ζ

⁢

(

-

n

,

K

)

is a

rational number

. On the other hand,

K

is obviously an

abelian number field

, thus the factorization of the Dedekind zeta function of an abelian number field tells us that:

ζ

⁢

(

s

,

K

)

=

ζ

⁢

(

s

)

⁢

L

⁢

(

s

,

χ

)

where

ζ

⁢

(

s

)

is the famous

Riemann zeta function

and

L

⁢

(

s

,

χ

)

is the

Dirichlet L-function

associated to

χ

, where

χ

is the unique

Dirichlet character

with

conductor

D

K

such that the group of

characters

of

K

/

ℚ

is

{

χ

0

,

χ

}

and

χ

0

is the trivial character. In fact, the values of

χ

are simply given by

χ

⁢

(

a

)

=

(

D

K

a

)

where the parentheses denote the

Kronecker symbol

.

Furthermore, if

k

is a

positive integer

then:

1.

Putting the values of the Riemann zeta function in terms of

Bernoulli numbers

one gets:

ζ

⁢

(

1

-

k

)

=

-

B

k

k

where

B

k

is the

k

th Bernoulli number;

2.

The values of Dirichlet L-series at negative integers can be written in terms of

generalized Bernoulli numbers

as follows:

L

⁢

(

1

-

k

,

χ

)

=

-

B

k

,

χ

k

where

B

k

,

χ

is the

k

th generalized Bernoulli number associated to

χ

.

Therefore:

ζ

⁢

(

1

-

k

,

K

)

=

ζ

⁢

(

1

-

k

)

⁢

L

⁢

(

1

-

k

,

χ

)

=

B

k

⋅

B

k

,

χ

k

2

.

The interested reader can find tables of values at the

http://www.math.cornell.edu/ alozano/dedekind-values/index.html

author’s personal website.

Title

values of Dedekind zeta functions of real quadratic number fields at negative integers

Canonical name

ValuesOfDedekindZetaFunctionsOfRealQuadraticNumberFieldsAtNegativeIntegers

Date of creation

2013-03-22 16:01:27

Last modified on

2013-03-22 16:01:27

Owner

alozano (2414)

Last modified by

alozano (2414)

Numerical id

5

Author

alozano (2414)

Entry type

Application

Classification

msc 11R42

Classification

msc 11M06

Related topic

FactorizationOfTheDedekindZetaFunctionOfAnAbelianNumberField

Generated on Fri Feb 9 15:45:25 2018 by

LaTeXML
values of the Dedekind zeta function of ℚ⁢(sqrt(5)) at negative integers

values of the Dedekind zeta function of

ℚ

⁢

(

5

)

at negative integers

As an example of the

http://planetmath.org/node/8064

parent entry, we exhibit a few values of the

Dedekind zeta function

ζ

⁢

(

s

,

K

)

associated to the

number field

K

=

ℚ

⁢

(

5

)

. The following are the first few values of

ζ

⁢

(

1

-

k

,

K

)

for

k

≥

2

even (

ζ

⁢

(

1

-

k

,

K

)

=

0

for odd

k

):

k

ζ

⁢

(

1

-

k

,

K

)

2

1

/

30

4

1

/

60

6

67

/

630

8

361

/

120

10

412751

/

1650

12

795286411

/

16380

14

568591843

/

30

16

54701427071177

/

4080

18

571363169189645713

/

35910

20

98510726938027364651

/

3300

22

58282448789678207092153

/

690

24

11364600197977872303826339891

/

32760

26

60097244486962154421889002337

/

30

28

27553534229181632149212403498558667

/

1740

30

179897691732312705009829008469165679351567

/

1074150

32

18959952687425212997198188226124948760804457

/

8160

34

1246857198273009466623646468891606914655865453

/

30

36

32613724432434114882079789497713006473885167100345949413

/

34545420

38

806046617423971060143933083079087480165364733020665319

/

30

40

256143677142597053855778958609943339670281438740737609308546491

/

270600

42

7744103059592598379932370402622990883812122603998003590194837592521

/

189630

44

2948486831915768046445018911155037347932306554653422623861453575795893

/

1380

46

189448047879825594335580602295205397993735968958688272066548049250586050109

/

1410

48

11220849734737403891785284646197712257525539376899788451604867938805644674521295667

/

1113840

50

7374830701193124787713243044624034212252297046043499899645907770420650321652018100351

/

8250

Title

values of the Dedekind zeta function of

ℚ

⁢

(

5

)

at negative integers

Canonical name

ValuesOfTheDedekindZetaFunctionOfmathbbQsqrt5AtNegativeIntegers

Date of creation

2013-03-22 16:01:29

Last modified on

2013-03-22 16:01:29

Owner

alozano (2414)

Last modified by

alozano (2414)

Numerical id

5

Author

alozano (2414)

Entry type

Example

Classification

msc 11R42

Classification

msc 11M06

Generated on Fri Feb 9 15:45:28 2018 by

LaTeXML
Mazur’s structure theorem

Mazur’s structure theorem

Any normed

associative

real

division algebra

is

isomorphic

to one of the following:

•

The

real numbers

ℝ

.

•

The

complex numbers

ℂ

.

•

The

quaternions

ℍ

.

The next

generalization

, the

octonions

, often viewed as the “complexification” of the quaternions, fails to be associative.

Title

Mazur’s

structure

theorem

Canonical name

MazursStructureTheorem

Date of creation

2013-03-22 14:50:04

Last modified on

2013-03-22 14:50:04

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

7

Author

mathcam (2727)

Entry type

Theorem

Classification

msc 11R52

Related topic

TheoremsOnSumsOfSquares

Generated on Fri Feb 9 15:45:29 2018 by

LaTeXML
quaternion algebra

quaternion algebra

A

quaternion algebra

over a field

K

is a

central simple algebra

over

K

which is four dimensional as a

vector space

over

K

.

Examples:

•

For any field

K

, the ring

M

2

×

2

⁢

(

K

)

of

2

×

2

matrices with entries in

K

is a quaternion algebra over

K

. If

K

is

algebraically closed

, then all quaternion algebras over

K

are

isomorphic

to

M

2

×

2

⁢

(

K

)

.

•

For

K

=

ℝ

, the well known

algebra

ℍ

of Hamiltonian quaternions is a quaternion algebra over

ℝ

. The two algebras

ℍ

and

M

2

×

2

⁢

(

ℝ

)

are the only quaternion algebras over

ℝ

, up to

isomorphism

.

•

When

K

is a

number field

, there are infinitely many non–isomorphic quaternion algebras over

K

. In fact, there is one such quaternion algebra for every even sized finite collection of

finite primes

or real primes of

K

. The proof of this deep fact leads to many of the major results of

class field theory

.

One can show that every quaternion algebra over

K

other than

M

2

×

2

⁢

(

K

)

is always a

division ring

.

Title

quaternion algebra

Canonical name

QuaternionAlgebra

Date of creation

2013-03-22 12:37:54

Last modified on

2013-03-22 12:37:54

Owner

djao (24)

Last modified by

djao (24)

Numerical id

5

Author

djao (24)

Entry type

Definition

Classification

msc 11R52

Classification

msc 16K20

Generated on Fri Feb 9 15:45:31 2018 by

LaTeXML
adèle

adèle

Let

K

be a

number field

. For each

finite prime

v

of

K

, let

𝔬

v

denote the

valuation ring

of the

completion

K

v

of

K

at

v

. The

adèle group

𝔸

K

of

K

is defined to be the

restricted direct product

of the collection of

locally compact

additive groups

{

K

v

}

over all primes

v

of

K

(both finite
primes and infinite primes), with respect to the collection of

compact

open subgroups

{

𝔬

v

}

defined for all finite primes

v

.

The set

𝔸

K

inherits

addition

and

multiplication

operations
(defined pointwise) which make it into a

topological ring

. The
original field

K

embeds as a ring into

𝔸

K

via the map

x

↦

∏

v

x

v

.

defined for

x

∈

K

, where

x

v

denotes the image of

x

in

K

v

under the

embedding

K

↪

K

v

. Note that

x

v

∈

𝔬

v

for all but finitely many

v

, so that the element

x

is sent under
the above definition into the restricted direct product as claimed.

It turns out that the image of

K

in

𝔸

K

is a

discrete set

and the

quotient group

𝔸

K

/

K

is a

compact space

in the

quotient topology

.

Title

adèle

Canonical name

Adele

Date of creation

2013-03-22 12:39:31

Last modified on

2013-03-22 12:39:31

Owner

djao (24)

Last modified by

djao (24)

Numerical id

5

Author

djao (24)

Entry type

Definition

Classification

msc 11R56

Related topic

Idele

Defines

adèle group

Defines

group of adèles

Generated on Fri Feb 9 15:45:33 2018 by

LaTeXML
grössencharacter

grössencharacter

Let

K

be a

number field

and let

A

K

be

idele group

of

K

, i.e.

A

K

=

∏

ν

K

ν

∗

where the

product

is a

restricted direct product

running

over all places (

infinite

and finite) of

K

(see entry on

http://planetmath.org/node/Idele

ideles). Recall that

K

∗

embeds into

A

K

diagonally:

x

∈

K

∗

↦

(

x

ν

)

ν

where

x

ν

is the image of

x

under the

embedding

of

K

into its

completion

at the place

ν

,

K

ν

.

Definition 1

.

A Grössencharacter

ψ

on

K

is a continuous

homomorphism

:

ψ

:

A

K

⟶

ℂ

∗

which is trivial on

K

∗

, i.e. if

x

∈

K

∗

then

ψ

⁢

(

(

x

ν

)

ν

)

=

1

. We say that

ψ

is

unramified

at a prime

℘

of

K

if

ψ

⁢

(

O

℘

∗

)

=

1

, where

O

℘

is the

ring of integers

inside

K

℘

. Otherwise we say that

ψ

is ramified at

℘

.

Let

𝒪

K

be the ring of integers in

K

. We may define a homomorphism on the (

multiplicative

) group of non-zero

fractional ideals

of

K

as follows. Let

℘

be a prime of

K

, let

π

be a

uniformizer

of

K

℘

and let

α

℘

∈

A

K

be the element which is

π

at the place

℘

and

1

at all other places. We define:

ψ

⁢

(

℘

)

=

{

0

,

if

⁢

ψ

⁢

is ramified at

⁢

℘

;

ψ

⁢

(

α

℘

)

,

otherwise

.

Definition 2

.

The Hecke L-series attached to a Grössencharacter

ψ

of

K

is given by the

Euler product

over all primes of

K

:

L

⁢

(

ψ

,

s

)

=

∏

℘

(

1

-

ψ

⁢

(

℘

)

(

N

ℚ

K

⁢

(

℘

)

)

s

)

-

1

.

Hecke L-series of this form have an analytic continuation and satisfy a certain functional equation. This fact was first proved by Hecke himself but later was vastly generalized by Tate using Fourier analysis on the ring

A

K

(what is usually called Tate’s thesis).

Title

grössencharacter

Canonical name

Grossencharacter

Date of creation

2013-03-22 15:45:19

Last modified on

2013-03-22 15:45:19

Owner

alozano (2414)

Last modified by

alozano (2414)

Numerical id

5

Author

alozano (2414)

Entry type

Definition

Classification

msc 11R56

Related topic

GrossencharacterAssociatedToACMEllipticCurve

Defines

grossencharacter

Generated on Fri Feb 9 15:45:38 2018 by

LaTeXML
idèle

idèle

Let

K

be a

number field

. For each

finite prime

v

of

K

, let

𝔬

v

be the

valuation ring

of the

completion

K

v

of

K

at

v

, and let

U

v

be the

group of units

in

𝔬

v

. Then each group

U

v

is a

compact

open subgroup

of the group of units

K

v

*

of

K

v

. The

idèle group

𝕀

K

of

K

is defined to be the

restricted direct product

of the

multiplicative groups

{

K

v

*

}

with respect to the compact open subgroups

{

U

v

}

, taken over all finite primes and infinite primes

v

of

K

.

The units

K

*

in

K

embed into

𝕀

K

via the

diagonal embedding

x

↦

∏

v

x

v

,

where

x

v

is the image of

x

under the

embedding

K

↪

K

v

of

K

into its completion

K

v

. As in the case of adèles, the group

K

*

is a discrete subgroup of the group of idèles

𝕀

K

, but unlike the case of adèles, the

quotient group

𝕀

K

/

K

*

is not a compact group. It is, however, possible to define a certain

subgroup

of the idèles (the subgroup of norm 1 elements) which does have compact

quotient

under

K

*

.

Warning:

The group

𝕀

K

is a

multiplicative

subgroup of the ring of adèles

𝔸

K

, but the

topology

on

𝕀

K

is different from the

subspace topology

that

𝕀

K

would have as a subset of

𝔸

K

.

Title

idèle

Canonical name

Idele

Date of creation

2013-03-22 12:39:28

Last modified on

2013-03-22 12:39:28

Owner

djao (24)

Last modified by

djao (24)

Numerical id

7

Author

djao (24)

Entry type

Definition

Classification

msc 11R56

Related topic

Adele

Defines

idèle group

Defines

group of idèles

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

LaTeXML
restricted direct product

restricted direct product

Let

{

G

v

}

v

∈

V

be a collection of

locally compact topological groups

. For all but finitely many

v

∈

V

, let

H

v

⊂

G

v

be a

compact

open subgroup

of

G

v

. The

restricted direct product

of the collection

{

G

v

}

with respect to the collection

{

H

v

}

is the

subgroup

G

:=

{

(

g

v

)

v

∈

V

∈

∏

v

∈

V

G

v

|

g

v

∈

H

v

⁢

for all but finitely many

v

∈

V

}

of the

direct product

∏

v

∈

V

G

v

.

We define a

topology

on

G

as follows. For every finite subset

S

⊂

V

that contains all the elements

v

for which

H

v

is undefined, form the

topological group

G

S

:=

∏

v

∈

S

G

v

×

∏

v

∉

S

H

v

consisting of the direct product of the

G

v

’s, for

v

∈

S

, and the

H

v

’s, for

v

∉

S

. The topological group

G

S

is a subset of

G

for each such

S

, and we take for a topology on

G

the weakest topology such that the

G

S

are open subsets of

G

, with the

subspace topology

on each

G

S

equal to the topology that

G

S

already has in its own right.

Title

restricted direct product

Canonical name

RestrictedDirectProduct

Date of creation

2013-03-22 12:35:38

Last modified on

2013-03-22 12:35:38

Owner

djao (24)

Last modified by

djao (24)

Numerical id

5

Author

djao (24)

Entry type

Definition

Classification

msc 11R56

Classification

msc 22D05

Generated on Fri Feb 9 15:45:42 2018 by

LaTeXML
Riemann-Hurwitz theorem

Riemann-Hurwitz theorem

First we define the different

divisor

of an

extension of function

fields. Let

K

be a

function field

over a field

F

and let

L

be a finite

separable extension

of

K

. Let

𝒪

P

be a prime of

K

, i.e. a

discrete valuation ring

with

F

⊂

𝒪

P

,

maximal ideal

P

and

quotient field

equal to

K

. Let

R

P

be the

integral closure

of

𝒪

P

in

L

. Notice that if

𝔭

is a

prime ideal

of

R

P

, then the

localization

𝒪

𝔭

=

(

R

P

)

𝔭

is a prime of

L

(which is said to be lying over

𝒪

P

). The maximal ideal of

𝒪

𝔭

is

𝔭

⁢

(

R

P

)

𝔭

.

Let

𝒪

𝔓

be any prime of

L

, then it lays over some prime ideal

P

of

K

and in fact, if

𝔭

=

R

P

∩

𝔓

then

𝒪

𝔭

≅

𝒪

𝔓

. Let

δ

⁢

(

𝔓

)

be the exact power of

𝔭

dividing the different of

R

P

over

𝒪

P

(the different of an

extension

of

Dedekind domains

is a

fractional ideal

). We define the different divisor of

L

/

K

as follows:

D

L

/

K

=

∑

𝔓

δ

⁢

(

𝔓

)

⁢

𝔓

as an element of the

free abelian group

generated by the prime ideals of

L

.

Theorem

(Riemann-Hurwitz).

Let

L

/

K

be a finite,

separable

, geometric extension of function fields and suppose the genus of

K

is

g

K

. Then the genus of

L

is given by the

formula

:

2

g

L

-

2

=

[

L

:

K

]

(

2

g

K

-

2

)

+

deg

L

D

L

/

K

.

Title

Riemann-Hurwitz theorem

Canonical name

RiemannHurwitzTheorem

Date of creation

2013-03-22 15:34:40

Last modified on

2013-03-22 15:34:40

Owner

alozano (2414)

Last modified by

alozano (2414)

Numerical id

4

Author

alozano (2414)

Entry type

Theorem

Classification

msc 11R58

Defines

different divisor of an extension of function fields

Generated on Fri Feb 9 15:45:48 2018 by

LaTeXML
cyclotomic polynomial

cyclotomic polynomial

\PMlinkescapephrase

root of unity

Definition

For any

positive integer

n

,
the

n

-th

cyclotomic polynomial

Φ

n

⁢

(

x

)

is defined as

Φ

n

⁢

(

x

)

=

∏

ζ

(

x

-

ζ

)

,

where

ζ

ranges over the
primitive

n

-th roots of unity (

http://planetmath.org/RootOfUnity

).

Examples

The first few cyclotomic polynomials are as follows:

Φ

1

⁢

(

x

)

=

x

-

1

Φ

2

⁢

(

x

)

=

x

+

1

Φ

3

⁢

(

x

)

=

x

2

+

x

+

1

Φ

4

⁢

(

x

)

=

x

2

+

1

Φ

5

⁢

(

x

)

=

x

4

+

x

3

+

x

2

+

x

+

1

Φ

6

⁢

(

x

)

=

x

2

-

x

+

1

Φ

7

⁢

(

x

)

=

x

6

+

x

5

+

x

4

+

x

3

+

x

2

+

x

+

1

Φ

8

⁢

(

x

)

=

x

4

+

1

Φ

9

⁢

(

x

)

=

x

6

+

x

3

+

1

Φ

10

⁢

(

x

)

=

x

4

-

x

3

+

x

2

-

x

+

1

Φ

11

⁢

(

x

)

=

x

10

+

x

9

+

x

8

+

x

7

+

x

6

+

x

5

+

x

4

+

x

3

+

x

2

+

x

+

1

Φ

12

⁢

(

x

)

=

x

4

-

x

2

+

1

The preceding examples may give the impression that the

coefficients

are always

-

1

,

0

or

1

, but this is not true in general.
For example,

Φ

105

⁢

(

x

)

=

x

48

+

x

47

+

x

46

-

x

43

-

x

42

-

2

⁢

x

41

-

x

40

-

x

39

+

x

36

+

x

35

+

x

34

+

x

33

+

x

32

+

x

31

-

x

28

-

x

26

-

x

24

-

x

22

-

x

20

+

x

17

+

x

16

+

x

15

+

x

14

+

x

13

+

x

12

-

x

9

-

x

8

-

2

⁢

x

7

-

x

6

-

x

5

+

x

2

+

x

+

1

Properties

For every positive integer

n

,

Φ

n

⁢

(

x

)

is an

irreducible polynomial

of degree

ϕ

⁢

(

n

)

in

ℚ

⁢

[

x

]

,
and is the

minimal polynomial

of each primitive

n

-th root of unity.
Here

ϕ

⁢

(

n

)

is Euler’s phi function.

Title

cyclotomic polynomial

Canonical name

CyclotomicPolynomial

Date of creation

2013-03-22 12:36:00

Last modified on

2013-03-22 12:36:00

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

14

Author

yark (2760)

Entry type

Definition

Classification

msc 11R60

Classification

msc 11R18

Classification

msc 11C08

Related topic

AllOnePolynomial

Related topic

FactoringAllOnePolynomialsUsingTheGroupingMethod

Related topic

CyclotomicField

Related topic

RootOfUnity

Generated on Fri Feb 9 15:45:51 2018 by

LaTeXML
factors of n and x^n-1

factors of

n

and

x

n

-

1

Let

n

be a

positive integer

.  Then the

binomial

x

n

-

1

has as many

prime factors

(

http://planetmath.org/PrimeFactorsOfXn1

) with

integer

coefficients

as the integer

n

has

positive

divisors

, both numbers thus being

τ

⁢

(

n

)

(

http://planetmath.org/TauFunction

).

Proof.

If

Φ

d

⁢

(

x

)

generally means the

d

th

cyclotomic polynomial

Φ

d

⁢

(

x

)

:=

(

x

-

ζ

1

)

⁢

(

x

-

ζ

2

)

⁢

…

⁢

(

x

-

ζ

φ

⁢

(

d

)

)

,

where the

ζ

j

s are the primitive

d

th

roots of unity

, then the equation

∏

d

⁢

|

n

,

d

>

⁢

0

Φ

d

⁢

(

x

)

=

x

n

-

1

is true, because each

n

th root of unity is also a primitive (

http://planetmath.org/RootOfUnity

)

d

th root of unity for one and only one positive divisor of

n

.  The cyclotomic factor polynomials

Φ

d

⁢

(

x

)

have integer coefficients and are

irreducible

(

http://planetmath.org/IrreduciblePolynomial2

).  Thus the number of them is same as the number

τ

⁢

(

n

)

of positive divisors of

n

.

For illustrating the proof, let

n

=

6

(divisors 1, 2, 3, 6); think the sixth roots of unity:

ζ

0

,

ζ

1

,

ζ

2

,

ζ

3

,

ζ

4

,

ζ

5

(where

ζ

=

e

i

⁢

π

/

3

=

1

+

i

⁢

3

2

).  From them,

ζ

0

=

1

is the primitive 1st root,

ζ

3

the primitive 2nd root,

ζ

2

and

ζ

4

the primitive 3rd roots,

ζ

1

and

ζ

5

the primitive 6th roots of unity.

Title

factors of

n

and

x

n

-

1

Canonical name

FactorsOfNAndXn1

Date of creation

2013-03-22 16:35:05

Last modified on

2013-03-22 16:35:05

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

7

Author

pahio (2872)

Entry type

Theorem

Classification

msc 11R60

Classification

msc 11C08

Classification

msc 11R18

Related topic

PrimeFactorsOfXn1

Generated on Fri Feb 9 15:45:59 2018 by

LaTeXML