aarjaneiro/planetmath huggingface dataset:
greatest common divisor

greatest common divisor

Let

a

and

b

be given

integers

, with at least one of them different from zero. The

greatest common divisor

of

a

and

b

, denoted by

gcd

⁡

(

a

,

b

)

, is the

positive integer

d

satisfying

1.

d

∣

a

and

d

∣

b

,

2.

if

c

∣

a

and

c

∣

b

, then

c

∣

d

.

More intuitively, the greatest common divisor is the largest integer dividing both

a

and

b

.

For example,

gcd

⁡

(

345

,

135

)

=

15

,

gcd

⁡

(

-

7

,

-

21

)

=

7

,

gcd

⁡

(

18

,

0

)

=

18

, and

gcd

⁡

(

44

,

97

)

=

1

Given two (

rational

) integers, one can construct their gcd via

Euclidean algorithm

. Once the gcd is found, it can be written as a

linear combination

of the two integers. That is, for any two integers

a

and

b

, we have

gcd

⁡

(

a

,

b

)

=

r

⁢

a

+

s

⁢

b

for some integers

r

and

s

. This

expression

is known as the

Bezout identity

.

One can generalize the notion of a gcd of two integers into a gcd of two

elements

in a commutative ring. However, given two elements in a general commutative ring, even an

integral domain

, a gcd may not exist. For example, in

ℤ

⁢

[

x

2

,

x

3

]

, a gcd for the elements

x

5

and

x

6

does not exist, for

x

2

and

x

3

are both common

divisors

of

x

5

and

x

6

, but neither one divides another.

The idea of the gcd of two integers can be generalized in another

direction

: to the gcd of a non-empty set of integers. If

S

is a non-empty set of integers, then the

gcd

of

S

, is a positive integer

d

such that

1.

d

∣

a

for all

a

∈

S

,

2.

if

c

∣

a

,

∀

a

∈

S

, then

c

∣

d

.

We denote

d

=

gcd

⁡

(

S

)

.

Remarks

.

•

gcd

⁡

(

a

,

b

,

c

)

=

gcd

⁡

(

gcd

⁡

(

a

,

b

)

,

c

)

.

•

If

∅

≠

T

⊆

S

⊆

ℤ

, then

gcd

(

S

)

∣

gcd

(

T

)

.

•

For any

∅

≠

S

⊆

ℤ

, there is a finite subset

T

⊆

S

such that

gcd

⁡

(

T

)

=

gcd

⁡

(

S

)

.

Title

greatest common divisor

Canonical

name

GreatestCommonDivisor

Date of creation

2013-03-22 11:46:50

Last modified on

2013-03-22 11:46:50

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

22

Author

CWoo (3771)

Entry type

Definition

Classification

msc 11-00

Classification

msc 03E20

Classification

msc 06F25

Synonym

gcd

Synonym

greatest common factor

Synonym

highest common factor

Synonym

hcf

Related topic

GcdDomain

Related topic

CorollaryOfBezoutsLemma

Related topic

ExampleOfGcd

Related topic

ExistenceAndUniquenessOfTheGcdOfTwoIntegers

Related topic

DivisionOfCongruence

Related topic

Congruences

Related topic

RationalSineAndCosine

Related topic

IntegralBasisOfQuadraticField

Related topic

SquareRootsOfRationals

Related topic

IntegerContraharmonicMeans

Related topic

SubgroupsWithCoprim

Generated on Fri Feb 9 15:07:16 2018 by

LaTeXML
Gregory series

Gregory series

The

Gregory series

is an

alternating sum

whose value is a quarter that of

π

:

π

4

=

∑

i

=

0

∞

(

-

1

)

i

⁢

1

2

⁢

i

+

1

=

1

-

1

3

+

1

5

-

1

7

+

1

9

-

…

(The approximate decimal value of this

expression

is 0.7853981633974483…)

More generally, a Gregory series for a given

n

is

∑

i

=

0

∞

(

-

1

)

i

⁢

n

2

⁢

i

+

1

2

⁢

i

+

1

.

The Gregory series is named after the Scottish astronomer and astrologer James Gregory.

Title

Gregory series

Canonical

name

GregorySeries

Date of creation

2013-03-22 17:35:33

Last modified on

2013-03-22 17:35:33

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

4

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 11-00

Classification

msc 51-00

Classification

msc 01A16

Classification

msc 01A20

Classification

msc 01A25

Classification

msc 01A32

Classification

msc 01A40

Related topic

TaylorSeriesOfArcusTangent

Generated on Fri Feb 9 15:07:18 2018 by

LaTeXML
group theoretic proof of Wilson’s theorem

group theoretic proof of Wilson’s theorem

Here we present a group theoretic proof of it.

Clearly, it is enough to show that

(

p

-

2

)

!

≡

1

(

mod

p

)

since

p

-

1

≡

-

1

(

mod

p

)

.
By

Sylow theorems

, we have that

p

-

Sylow subgroups

of

S

p

, the

symmetric group

on

p

elements, have order

p

, and the number

n

p

of
Sylow subgroups is congruent to 1 modulo

p

. Let

P

be a Sylow subgroup
of

S

p

. Note that

P

is generated by a

p

-cycle. There are

(

p

-

1

)

!

cycles
of length

p

in

S

p

. Each

p

-Sylow subgroup contains

p

-

1

cycles
of length

p

, hence there are

(

p

-

1

)

!

p

-

1

=

(

p

-

2

)

!

different

p

-Sylow subgrups in

S

p

, i.e.

n

P

=

(

p

-

2

)

!

. From Sylow’s Second
Theorem, it follows that

(

p

-

2

)

!

≡

1

(

mod

p

)

,so

(

p

-

1

)

!

≡

-

1

(

mod

p

)

.

Title

group theoretic proof of Wilson’s theorem

Canonical name

GroupTheoreticProofOfWilsonsTheorem

Date of creation

2013-03-22 13:35:27

Last modified on

2013-03-22 13:35:27

Owner

ottocolori (1519)

Last modified by

ottocolori (1519)

Numerical id

10

Author

ottocolori (1519)

Entry type

Proof

Classification

msc 11-00

Generated on Fri Feb 9 15:07:20 2018 by

LaTeXML
harmonic mean

harmonic mean

If

a

1

,

a

2

,

…

,

a

n

are positive numbers, we define their

harmonic mean

as the

inverse number

of the

arithmetic mean

of their inverse numbers:

H

.

M

.

=

n

1

a

1

+

1

a

2

+

⋯

+

1

a

n

•

It follows easily the estimation

H

.

M

.

<

n

a

i

(

i

=

1

,

2

,

…

,

n

)

.

•

If you travel from city

A

to city

B

at

x

miles per hour, and then you travel back at

y

miles per hour.  What was the

average

velocity for the whole trip?

The harmonic mean of

x

and

y

. That is, the average velocity is

2

1

x

+

1

y

=

2

⁢

x

⁢

y

x

+

y

.

•

If one draws through the intersecting point of the

diagonals

of a

trapezoid

a line

parallel

to the parallel sides of the trapezoid, then the

segment

of the line inside the trapezoid is equal to the harmonic mean of the parallel sides.

•

In the harmonic series

1

+

1

2

+

1

3

+

1

4

+

…

every following it.

Title

harmonic mean

Canonical name

HarmonicMean

Date of creation

2013-03-22 11:50:50

Last modified on

2013-03-22 11:50:50

Owner

drini (3)

Last modified by

drini (3)

Numerical id

15

Author

drini (3)

Entry type

Definition

Classification

msc 11-00

Related topic

ArithmeticMean

Related topic

GeneralMeansInequality

Related topic

WeightedPowerMean

Related topic

PowerMean

Related topic

ArithmeticGeometricMeansInequality

Related topic

RootMeanSquare3

Related topic

ProofOfGeneralMeansInequality

Related topic

ProofOfArithmeticGeometricHarmonicMeansInequality

Related topic

HarmonicMeanInTrapezoid

Related topic

ContraharmonicMean

Related topic

Contr

Generated on Fri Feb 9 15:07:22 2018 by

LaTeXML
index of an integer with respect to a primitive root

index of an integer with respect to a primitive root

Definition

.

Let

m

>

1

be an

integer

such that the integer

g

is a

primitive root

for

m

. Suppose

a

is another integer relatively prime to

g

. The index of

a

(to base

g

) is the smallest

positive integer

n

such that

g

n

≡

a

mod

m

, and it is denoted by

ind

⁡

a

or

ind

g

⁡

a

.

If

m

has a primitive root the index with respect to a primitive root is a very useful tool to solve

polynomial congruences

modulo

m

.

Title

index of an integer with respect to a primitive root

Canonical name

IndexOfAnIntegerWithRespectToAPrimitiveRoot

Date of creation

2013-03-22 16:20:50

Last modified on

2013-03-22 16:20:50

Owner

alozano (2414)

Last modified by

alozano (2414)

Numerical id

4

Author

alozano (2414)

Entry type

Definition

Classification

msc 11-00

Generated on Fri Feb 9 15:07:23 2018 by

LaTeXML
integer

integer

The set of integers, denoted by the symbol

ℤ

, is the set

{

…

-

3

,

-

2

,

-

1

,

0

,

1

,

2

,

3

,

…

}

consisting of the

natural numbers

and their

negatives

.

Mathematically,

ℤ

is defined to be the set of

equivalence classes

of pairs of natural numbers

ℕ

×

ℕ

under the

equivalence relation

(

a

,

b

)

∼

(

c

,

d

)

if

a

+

d

=

b

+

c

.

Addition

and

multiplication

of

integers

are defined as follows:

•

(

a

,

b

)

+

(

c

,

d

)

:=

(

a

+

c

,

b

+

d

)

•

(

a

,

b

)

⋅

(

c

,

d

)

:=

(

a

⁢

c

+

b

⁢

d

,

a

⁢

d

+

b

⁢

c

)

Typically, the class of

(

a

,

b

)

is denoted by symbol

n

if

b

≤

a

(resp.

-

n

if

a

≤

b

), where

n

is the unique natural number such that

a

=

b

+

n

(resp.

a

+

n

=

b

). Under this notation, we recover the familiar representation of the integers as

{

…

,

-

3

,

-

2

,

-

1

,

0

,

1

,

2

,

3

,

…

}

. Here are some examples:

•

0

=

equivalence class of

(

0

,

0

)

=

equivalence class of

(

1

,

1

)

=

…

•

1

=

equivalence class of

(

1

,

0

)

=

equivalence class of

(

2

,

1

)

=

…

•

-

1

=

equivalence class of

(

0

,

1

)

=

equivalence class of

(

1

,

2

)

=

…

The set of integers

ℤ

under the addition and multiplication

operations

defined above form an

integral domain

. The integers admit the following

ordering relation

making

ℤ

into an

ordered ring

:

(

a

,

b

)

≤

(

c

,

d

)

in

ℤ

if

a

+

d

≤

b

+

c

in

ℕ

.

The

ring of integers

is also a

Euclidean domain

, with

valuation

given by the

absolute value

function

.

Title

integer

Canonical name

Integer

Date of creation

2013-03-22 11:50:39

Last modified on

2013-03-22 11:50:39

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

13

Author

CWoo (3771)

Entry type

Definition

Classification

msc 11-00

Classification

msc 03-00

Synonym

rational integer

Synonym

ℤ

Related topic

Irrational

Generated on Fri Feb 9 15:07:25 2018 by

LaTeXML
integer part

integer part

The

integer part

of a

real number

is the part of the number that appears before the decimal . For example, the integer part of

π

is

3

, and the integer part of

-

2

is

-

1

.

To be more precise, for

x

∈

ℝ

, the integer part of

x

, denoted as

[

x

]

, is given by

[

x

]

=

{

⌊

x

⌋

⁢

if

⁢

x

≥

0

⌈

x

⌉

⁢

if

⁢

x

<

0

,

where

⌊

x

⌋

and

⌈

x

⌉

denote the floor and

ceiling

of

x

, respectively.

Title

integer part

Canonical

name

IntegerPart

Date of creation

2013-03-22 16:14:11

Last modified on

2013-03-22 16:14:11

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

6

Author

Wkbj79 (1863)

Entry type

Definition

Classification

msc 11-00

Classification

msc 26A09

Related topic

FractionalPart

Generated on Fri Feb 9 15:07:28 2018 by

LaTeXML
Jordan’s totient function

Jordan’s totient function

Let

p

be a prime,

k

and

n

natural numbers

. Then

J

k

⁢

(

n

)

=

n

k

⁢

∏

p

|

n

(

1

-

p

-

k

)

where the

product

is over

divisors

of

n

.

This is a

generalization

of Euler’s

Totient Function

.

Title

Jordan’s totient function

Canonical name

JordansTotientFunction

Date of creation

2013-03-22 11:42:21

Last modified on

2013-03-22 11:42:21

Owner

akrowne (2)

Last modified by

akrowne (2)

Numerical id

39

Author

akrowne (2)

Entry type

Definition

Classification

msc 11-00

Classification

msc 46M15

Classification

msc 18C15

Generated on Fri Feb 9 15:07:29 2018 by

LaTeXML
least common multiple

least common multiple

If

a

and

b

are two

positive integers

, then their

least
common

multiple

, denoted by

lcm

⁢

(

a

,

b

)

,

is the positive
integer

f

satisfying the conditions

•

a

∣

f

and

b

∣

f

,

•

if

a

∣

c

and

b

∣

c

, then

f

∣

c

.

Note:

The definition can be generalized for several
numbers.  The

positive

lcm of positive integers is
uniquely determined. (Its negative satisfies the same two
conditions.)

Properties

1.

If

a

=

∏

i

=

1

m

p

i

α

i

and

b

=

∏

i

=

1

m

p

i

β

i

are the

prime factor

of the positive integers

a

and

b

(

α

i

≧

0

,

β

i

≧

0

∀

i

), then

lcm

⁢

(

a

,

b

)

=

∏

i

=

1

m

p

i

max

⁡

{

α

i

,

β

i

}

.

This can be generalized for lcm of several numbers.

2.

Because the

greatest common divisor

has the expression

gcd

⁡

(

a

,

b

)

=

∏

i

=

1

m

p

i

min

⁡

{

α

i

,

β

i

}

, we see that

gcd

⁡

(

a

,

b

)

⋅

lcm

⁢

(

a

,

b

)

=

a

⁢

b

.

This formula is sensible only for two

integers

; it can not be
generalized for several numbers, i.e., for example,

gcd

⁡

(

a

,

b

,

c

)

⋅

lcm

⁢

(

a

,

b

,

c

)

≠

a

⁢

b

⁢

c

.

3.

The preceding formula may be presented in
of ideals of

ℤ

; we may
replace the integers with the corresponding

principal ideals

.
The formula acquires the form

(

(

a

)

+

(

b

)

)

⁢

(

(

a

)

∩

(

b

)

)

=

(

a

)

⁢

(

b

)

.

4.

The recent formula is valid also for other than principal ideals and even in so general systems as the Prüfer rings; in fact, it could be taken as defining property of these rings:   Let

R

be a

commutative ring

with

non-zero unity

.

R

is a Prüfer ring iff

Jensen’s formula

(

𝔞

+

𝔟

)

⁢

(

𝔞

∩

𝔟

)

=

𝔞

⁢

𝔟

is true for all ideals

𝔞

and

𝔟

of

R

, with at least one of them having non-zero-divisors (

http://planetmath.org/ZeroDivisor

).

References

1

M. Larsen and P. McCarthy:

Multiplicative

theory of ideals

. Academic Press. New York (1971).

Title

least common multiple

Canonical name

LeastCommonMultiple

Date of creation

2015-05-06 19:07:25

Last modified on

2015-05-06 19:07:25

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

32

Author

pahio (2872)

Entry type

Definition

Classification

msc 11-00

Synonym

least common

dividend

Synonym

lcm

Related topic

Divisibility

Related topic

PruferRing

Related topic

SumOfIdeals

Related topic

IdealOfElementsWithFiniteOrder

Generated on Fri Feb 9 15:07:31 2018 by

LaTeXML
Legendre symbol

Legendre symbol

Legendre Symbol

.

Let

p

be an

odd prime

. The

Legendre symbol

(

a

p

)

or

(

a

|

p

)

is defined as:

(

a

p

)

=

{

1

if

⁢

a

⁢

is a quadratic residue

(

mod

p

)

-

1

if

⁢

a

⁢

is a quadratic nonresidue

(

mod

p

)

0

if

⁢

p

⁢

divides

⁢

a

The Legendre symbol can be computed by means of Euler’s criterion or Gauss’ lemma.

Generalizations

of this symbol are the

Jacobi Symbol

and the

Kronecker symbol

.

Title

Legendre symbol

Canonical name

LegendreSymbol

Date of creation

2013-03-22 11:44:46

Last modified on

2013-03-22 11:44:46

Owner

alozano (2414)

Last modified by

alozano (2414)

Numerical id

15

Author

alozano (2414)

Entry type

Definition

Classification

msc 11-00

Classification

msc 97U20

Related topic

JacobiSymbol

Related topic

EulersCriterion

Related topic

QuadraticResidue

Related topic

KroneckerSymbol

Related topic

QuadraticReciprocityRule

Related topic

QuadraticCongruence

Generated on Fri Feb 9 15:07:33 2018 by

LaTeXML
linear formulas for Pythagorean triples

linear formulas for Pythagorean triples

It is easy to see that the

equation

a

2

+

b

2

=

c

2

(1)

of the

Pythagorean theorem

(

http://planetmath.org/PythagorasTheorem

) is

equivalent

(

http://planetmath.org/Equivalent3

) with

(

a

+

b

-

c

)

2

=

2

⁢

(

c

-

a

)

⁢

(

c

-

b

)

.

(2)

When

(

a

,

b

,

c

)

is a

Pythagorean triple

, i.e.

a

,

b

,

c

are

positive integers

,

a

+

b

-

c

must be an even positive
integer which we denote by

2

⁢

r

.  We get from (2) the equation

(

c

-

a

)

⁢

(

c

-

b

)

=

2

⁢

r

2

,

whose factors (

http://planetmath.org/

Product

) on the left hand side we denote
by

t

and

s

.  Thus we have the linear equation system

{

a

+

b

-

c

=

2

⁢

r

,

c

-

a

=

t

,

c

-

b

=

s

.

Its solution is

{

a

=

2

⁢

r

+

s

,

b

=

2

⁢

r

+

t

,

c

=

2

⁢

r

+

s

+

t

.

(3)

Here,

r

is an arbitrary positive integer,

s

and

t

are two
positive integers whose product is

2

⁢

r

2

.  It’s clear that then
(3) produces all Pythagorean triples.

References

1

Egon Scheffold

: “Ein Bild der pythagoreischen
Zahlentripel”.  –

Elemente der Mathematik

50

(1995).

Title

linear

formulas

for Pythagorean triples

Canonical name

LinearFormulasForPythagoreanTriples

Date of creation

2014-12-22 21:59:51

Last modified on

2014-12-22 21:59:51

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

13

Author

pahio (2872)

Entry type

Result

Classification

msc 11-00

Related topic

DerivationOfPythagoreanTriples

Related topic

ContraharmonicMeansAndPythagoreanHypotenuses

Related topic

DeterminingIntegerContraharmonicMeans

Generated on Fri Feb 9 15:07:42 2018 by

LaTeXML
mean

mean

Loosely speaking, a mean is a way to describe a

collection

of numbers such
that the mean in some sense describe the “

average

” entry of these numbers.
The most familiar mean is the

arithmetic mean

, and unless otherwise noted, by mean,
we always mean the arithmetic mean.

Example

The mean of the numbers

{

1

,

2

,

…

,

n

}

is

n

+

1

2

.

Mathematically, we define a mean as follows:

Definition

A

mean

is a

function

f

whose domain is the collection of
all finite

multisets

of

ℝ

and whose

codomain

is

ℝ

,
such that

•

f

is a homogeneous function of degree 1.  That is, if

{

x

1

,

…

,

x

n

}

is a multiset, then

f

⁢

(

{

λ

⁢

x

1

,

…

,

λ

⁢

x

n

}

)

=

λ

⁢

f

⁢

(

{

x

1

,

…

,

x

n

}

)

,

λ

≥

0

.

•

For any set

S

=

{

x

1

,

…

,

x

n

}

of

real numbers

,

min

⁡

{

x

1

,

…

,

x

n

}

≤

f

⁢

(

S

)

≤

max

⁡

{

x

1

,

…

,

x

n

}

.

Pythagoras identified three types of means: the arithmetic mean (

http://

planetmath

.org/ArithmeticMean

), the

geometric
mean

, and the

harmonic mean

. However, in the sense of the above

definition

,
there is a wealth of ther means too. For

instance

, the

minimum

function and

maximum

functions can be seen as “

trivial

” means. Other well-known means include:

•

median,

•

mode,

•

generalized mean

•

power mean

•

Lehmer mean

•

arithmetic-geometric mean

,

•

arithmetic-harmonic mean

,

•

harmonic-geometric mean

,

•

root-mean-square

(sometimes called the quadratic mean),

•

identric mean

,

•

contraharmonic mean

,

•

Heronian mean

,

•

Cesaro mean,

•

maximum function, minimum function (

http://planetmath.org/MinimalAndMaximalNumber

)

Title

mean

Canonical

name

Mean

Date of creation

2013-03-22 12:43:43

Last modified on

2013-03-22 12:43:43

Owner

matte (1858)

Last modified by

matte (1858)

Numerical id

16

Author

matte (1858)

Entry type

Definition

Classification

msc 11-00

Classification

msc 62-07

Related topic

ArithmeticMean

Related topic

GeometricMean

Related topic

ContraharmonicProportion

Related topic

OrderOfSixMeans

Related topic

AverageValueOfFunction

Generated on Fri Feb 9 15:07:52 2018 by

LaTeXML
mutually coprime

mutually coprime

Natural numbers

a

1

,

a

2

,

…

,

a

n

are called

mutually coprime

if there is no

p

∈

ℕ

,

p

>

1

with

p

|

a

i

for all

i

≤

n

. By this definition for example

15

,

21

and

25

would be mutually coprime though they are not

pairwise coprime

, because that would that each pair of these numbers is

coprime

.

Title

mutually coprime

Canonical name

MutuallyCoprime

Date of creation

2013-03-22 14:18:07

Last modified on

2013-03-22 14:18:07

Owner

mathwizard (128)

Last modified by

mathwizard (128)

Numerical id

5

Author

mathwizard (128)

Entry type

Definition

Classification

msc 11-00

Defines

pairwise coprime

Generated on Fri Feb 9 15:07:54 2018 by

LaTeXML
number field

number field

Definition 1

.

A field which is a

finite extension

of

Q

, the

rational numbers

, is called a

number field

(sometimes called

algebraic number field

). If the degree of the

extension

K

/

Q

is

n

then we say that

K

is a number field of degree

n

(over

Q

).

Example 1

.

The

field of rational

numbers

ℚ

is a number field.

Example 2

.

Let

K

=

ℚ

⁢

(

d

)

, where

d

≠

1

is a

square-free

non-zero

integer

and

d

stands for any of the roots of

x

2

-

d

=

0

(note that if

d

∈

K

then

-

d

∈

K

as well). Then

K

is a number field and

[

K

:

ℚ

]

=

2

. We can explictly describe all elements of

K

as follows:

K

=

{

t

+

s

⁢

d

:

t

,

s

∈

ℚ

}

.

Definition 2

.

A number field

K

such that the degree of the extension

K

/

Q

is

2

is called a

quadratic number field

.

In fact, if

K

is a quadratic number field, then it is easy to show that

K

is one of the fields described in Example

2

.

Example 3

.

Let

K

n

=

ℚ

⁢

(

ζ

n

)

be a

cyclotomic extension

of

ℚ

, where

ζ

n

is a

primitive

n

th

root of unity

. Then

K

is a number field and

[

K

:

ℚ

]

=

φ

(

n

)

where

φ

⁢

(

n

)

is the

Euler phi function

. In particular,

φ

⁢

(

3

)

=

2

, therefore

K

3

is a quadratic number field (in fact

K

3

=

ℚ

⁢

(

-

3

)

). We can explicitly describe all elements of

K

as follows:

K

n

=

{

q

0

+

q

1

⁢

ζ

n

+

q

2

⁢

ζ

n

2

+

…

+

q

n

-

1

⁢

ζ

n

n

-

1

:

q

i

∈

ℚ

}

.

In fact, one can do better. Every element of

K

n

can be uniquely expressed as a rational

combination

of the

φ

⁢

(

n

)

elements

{

ζ

n

a

:

gcd

⁡

(

a

,

n

)

=

1

,

1

≤

a

<

n

}

.

Example 4

.

Let

K

be a number field. Then any

subfield

L

with

ℚ

⊆

L

⊆

K

is also a number field. For example, let

p

be a

prime number

and let

F

=

ℚ

⁢

(

ζ

p

)

be a cyclotomic extension of

ℚ

, where

ζ

p

is a primitive

p

th root of unity. Let

F

+

be the

maximal

real subfield of

F

.

F

+

is a number field and it can be shown that:

F

+

=

ℚ

⁢

(

ζ

p

+

ζ

p

-

1

)

.

Title

number field

Canonical name

NumberField

Date of creation

2013-03-22 12:04:09

Last modified on

2013-03-22 12:04:09

Owner

alozano (2414)

Last modified by

alozano (2414)

Numerical id

17

Author

alozano (2414)

Entry type

Definition

Classification

msc 11-00

Synonym

algebraic number field

Related topic

AlgebraicNumberTheory

Related topic

ExamplesOfPrimeIdealDecompositionInNumberFields

Related topic

ExamplesOfFields

Related topic

AbelianExtensionsOfQuadraticImaginaryNumberFields

Related topic

NumberTheory

Related topic

ResidueDegree

Related topic

Regulator

Related topic

DiscriminantIdeal

Related topic

ClassNumber2

Related topic

ExistenceOfHilbertClassField

Related topic

Multiplicat

Defines

quadratic number field

Defines

quadratic field

Generated on Fri Feb 9 15:07:56 2018 by

LaTeXML
primitive root

primitive root

Given any

positive integer

n

, the

group of units

U

⁢

(

ℤ

/

n

⁢

ℤ

)

of the ring

ℤ

/

n

⁢

ℤ

is a

cyclic group

iff

n

is 4,

p

m

or

2

⁢

p

m

for any odd

positive

prime

p

and any non-negative

integer

m

.  A

primitive root

is a

generator

of this group of units when it is cyclic.

Equivalently, one can define the integer

r

to be a

primitive root modulo

n

, if the numbers

r

0

,

r

1

,

…

,

r

n

-

2

form a

reduced residue system

modulo

n

.

For example, 2 is a primitive root modulo 5, since

1

,

2

,

2

2

=

4

,

2

3

=

8

≡

3

(

mod

5

)

are all with 5

coprime

positive integers less than 5.

The

generalized Riemann hypothesis

implies that every

prime number

p

has a primitive root below

70

⁢

(

ln

⁡

p

)

2

.

References

Wikipedia, “Primitive root modulo n”

Title

primitive root

Canonical name

PrimitiveRoot

Date of creation

2013-03-22 16:04:33

Last modified on

2013-03-22 16:04:33

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

12

Author

CWoo (3771)

Entry type

Definition

Classification

msc 11-00

Synonym

primitive root modulo n

Synonym

primitive element

Related topic

MultiplicativeOrderOfAnIntegerModuloM

Related topic

PrimeResidueClass

Related topic

UsingPrimitiveRootsAndIndexToSolveCongruences

Generated on Fri Feb 9 15:07:57 2018 by

LaTeXML