aarjaneiro/planetmath huggingface dataset:
classification of complex numbers

classification of complex numbers

The set

ℂ

of all

complex numbers

and many of its subsets may be
partitioned (classified) into two subsets by certain criterion of the numbers.

A.

F i r s t   c l a s s i f i c a t i o n :

Complex numbers

contain

1.

algebraic numbers

2.

transcendental numbers

Algebraic numbers

contain

1.

algebraic integers

( algebraic numbers)

2.

algebraic fractions (fractional algebraic numbers)

Algebraic integers

contain

1.

rational integers

2.

non-rational

integers

Algebraic fractions

contain

1.

rational

fractions

2.

non-rational fractions

Transcendental numbers

contain

1.

real transcendental numbers

2.

imaginary

transcendental numbers

B.

S e c o n d   c l a s s i f i c a t i o n :

Complex numbers

contain

1.

real numbers

(

http://

planetmath

.org/RealNumber

) (the set

ℝ

)

2.

imaginary numbers

(i.e. non-real complex numbers)

Real numbers

contain

1.

rational numbers

(the set

ℚ

)

2.

irrational numbers

Rational numbers

contain

1.

integers (

http://planetmath.org/Integer

) (the set

ℤ

)

2.

fractional numbers

Imaginary numbers

contain

1.

pure imaginary numbers (with

real part

0)

2.

other imaginary numbers (with real part

≠

0

)

One can also combine the criterions of

A

and

B

; thus e.g.
the irrational numbers consist of the

algebraic

irrational numbers and
the irrational numbers.

In , any of the sets

ℝ

,

ℚ

and

ℤ

may be partitioneded into positive numbers, negative numbers and 0 (

http://planetmath.org/Null

).

Number-theoretically, the set

ℤ

consists of four of integers:

1

o

the number 0,

2

o

the units of

ℤ

(only

+

1

and

-

1

),

3

o

the

prime numbers

(

±

2

,

±

3

,

±

5

,

±

7

,

±

11

,

…

),

4

o

the

composite numbers

(

±

4

,

±

6

,

±

8

,

±

9

,

±

10

,

…

)

Title

classification of complex numbers

Canonical name

ClassificationOfComplexNumbers

Date of creation

2013-03-22 16:56:49

Last modified on

2013-03-22 16:56:49

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

11

Author

pahio (2872)

Entry type

Topic

Classification

msc 11R04

Related topic

NegativeNumber

Related topic

Number

Generated on Fri Feb 9 15:41:00 2018 by

LaTeXML
condition for power basis

condition for power basis

Lemma.

If

K

is an

algebraic number field

of degree (

http://planetmath.org/Degree

)

n

and the elements

α

1

,

α

2

,

…

,

α

n

of

K

can be expressed as

linear combinations

{

α

1

=

c

11

⁢

β

1

+

c

12

⁢

β

2

+

…

+

c

1

⁢

n

⁢

β

n

α

2

=

c

21

⁢

β

1

+

c

22

⁢

β

2

+

…

+

c

2

⁢

n

⁢

β

n

⋯

α

n

=

c

n

⁢

1

⁢

β

1

+

c

n

⁢

2

⁢

β

2

+

…

+

c

n

⁢

n

⁢

β

n

of the elements

β

1

,

β

2

,

…

,

β

n

of

K

with

rational

coefficients

c

i

⁢

j

, then the

discriminants

of

α

i

and

β

j

are by the equation

Δ

⁢

(

α

1

,

α

2

,

…

,

α

n

)

=

det

⁡

(

c

i

⁢

j

)

2

⋅

Δ

⁢

(

β

1

,

β

2

,

…

,

β

n

)

.

Theorem.

Let

ϑ

be an

algebraic integer

of degree (

http://planetmath.org/DegreeOfAnAlgebraicNumber

)

n

.  The set

{

1

,

ϑ

,

…

,

ϑ

n

-

1

}

is an

integral basis

of

ℚ

⁢

(

ϑ

)

if the discriminant

d

⁢

(

ϑ

)

:=

Δ

⁢

(

1

,

ϑ

,

…

,

ϑ

n

-

1

)

is

square-free

.

Proof.

The

adjusted canonical basis

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

of

ℚ

⁢

(

ϑ

)

is an integral basis, where

d

2

,

d

3

,

…

,

d

n

are

integers

.  Its discriminant is the

fundamental number

d

of the field.  By the lemma, we obtain

d

=

Δ

⁢

(

ω

1

,

ω

2

,

…

,

ω

n

)

=

|

1

0

…

0

a

21

d

2

1

d

2

⋱

0

⋮

⋮

⋱

0

a

n

⁢

1

d

n

a

n

⁢

2

d

n

…

1

d

n

|

2

⁢

Δ

⁢

(

1

,

ϑ

,

…

,

ϑ

n

-

1

)

=

d

⁢

(

ϑ

)

(

d

2

⁢

d

3

⁢

⋯

⁢

d

n

)

2

.

Thus

(

d

2

⁢

d

3

⁢

⋯

⁢

d

n

)

2

⁢

d

=

d

⁢

(

ϑ

)

,  and since

d

⁢

(

ϑ

)

is assumed to be square-free, we have

(

d

2

⁢

d

3

⁢

⋯

⁢

d

n

)

2

=

1

,  and accordingly

d

⁢

(

ϑ

)

equals the discriminant of the field (

http://planetmath.org/MinimalityOfIntegralBasis

).  This implies (see minimality of integral basis) that the numbers

1

,

ϑ

,

…

,

ϑ

n

-

1

form an integral basis of the field

ℚ

⁢

(

ϑ

)

.

Title

condition for power basis

Canonical name

ConditionForPowerBasis

Date of creation

2013-03-22 17:49:56

Last modified on

2013-03-22 17:49:56

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

9

Author

pahio (2872)

Entry type

Theorem

Classification

msc 11R04

Related topic

IntegralBasis

Related topic

PowerBasis

Related topic

CanonicalBasis

Related topic

PropertiesOfDiscriminantInAlgebraicNumberField

Generated on Fri Feb 9 15:41:04 2018 by

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Dirichlet’s unit theorem

Dirichlet’s unit theorem

\PMlinkescapephrase

occur in

Let

K

be a

number field

, and let

𝒪

K

be its

ring of integers

.
Then

𝒪

K

*

≅

μ

⁢

(

K

)

×

ℤ

r

+

s

-

1

.

Here

𝒪

K

*

is the

group of units

of

𝒪

K

,

μ

⁢

(

K

)

is the finite cyclic group of the

roots of unity

in

𝒪

K

*

,

r

is the number of

real embeddings

K

→

ℝ

,
and

2

⁢

s

is the number of non-real complex embeddings

K

→

ℂ

(which occur in

complex conjugate

pairs, so

s

is an

integer

).

Title

Dirichlet’s unit theorem

Canonical name

DirichletsUnitTheorem

Date of creation

2013-03-22 13:22:42

Last modified on

2013-03-22 13:22:42

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

10

Author

yark (2760)

Entry type

Theorem

Classification

msc 11R04

Classification

msc 11R27

Related topic

Regulator

Generated on Fri Feb 9 15:41:09 2018 by

LaTeXML
Eisenstein integers

Eisenstein integers

Let

ρ

=

(

-

1

+

-

3

)

/

2

, where we arbitrarily choose

-

3

to be either of the

complex numbers

whose square is

-

3

. Note that

ρ

3

=

1

. The

Eisenstein integers

are the ring

ℤ

⁢

[

ρ

]

=

{

a

+

b

⁢

ρ

:

a

,

b

∈

ℤ

}

.

Title

Eisenstein integers

Canonical

name

EisensteinIntegers

Date of creation

2013-03-22 11:45:35

Last modified on

2013-03-22 11:45:35

Owner

KimJ (5)

Last modified by

KimJ (5)

Numerical id

13

Author

KimJ (5)

Entry type

Definition

Classification

msc 11R04

Classification

msc 65A05

Classification

msc 65D20

Classification

msc 00-01

Classification

msc 11Y70

Classification

msc 18-00

Related topic

GaussianIntegers

Related topic

ComplexNumber

Related topic

NumberField

Related topic

EisenteinPrime

Generated on Fri Feb 9 15:41:10 2018 by

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Eisenstein prime

Eisenstein prime

Given the

complex

cubic

root of unity

ω

=

e

2

⁢

i

⁢

π

3

, an

Eisenstein integer

a

⁢

ω

+

b

(where

a

and

b

are natural

integers

) is said to be an

Eisenstein prime

if its only

divisors

are 1,

ω

,

1

+

ω

and itself.

Eisenstein primes of the form

0

⁢

ω

+

b

are ordinary natural primes

p

≡

2

mod

3

. Therefore no

Mersenne prime

is also an Eisenstein prime.

Title

Eisenstein prime

Canonical

name

EisensteinPrime

Date of creation

2013-03-22 16:10:10

Last modified on

2013-03-22 16:10:10

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

6

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 11R04

Related topic

EisensteinIntegers

Generated on Fri Feb 9 15:41:12 2018 by

LaTeXML
examples of characteristic functions

examples of characteristic functions

\PMlinkescapephrase

closed form

In this entry, we give some

examples of characteristic functions

written in closed form (i.e. (

http://planetmath.org/Ie

), not

piecewise

defined).

We use the following conventions:

•

X

denotes the domain (

http://planetmath.org/Domain

) of the

characteristic function

•

A

denotes a subset of

X

•

χ

A

denotes the characteristic function of

A

•

|

⋅

|

denotes distance from

0

(e.g. (

http://planetmath.org/Eg

)

absolute value

or

modulus

(

http://planetmath.org/Modulus2

))

•

⌊

⋅

⌋

denotes the

floor function

•

⌈

⋅

⌉

denotes the

ceiling function

•

Re

denotes the

real part

of a

complex number

•

Im

denotes the

imaginary part

of a complex number

•

ℕ

denotes the

positive integers

•

μ

denotes the Möbius

function

•

φ

denotes the

Euler totient function

•

𝒪

K

denotes the

ring of integers

of a

number field

K

•

N

K

denotes the norm on a number field

K

Below is a of characteristic functions:

Title

examples of characteristic functions

Canonical name

ExamplesOfCharacteristicFunctions

Date of creation

2013-03-22 17:54:28

Last modified on

2013-03-22 17:54:28

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

11

Author

Wkbj79 (1863)

Entry type

Example

Classification

msc 11R04

Classification

msc 11A25

Classification

msc 26A09

Classification

msc 26-00

\@unrecurse

Generated on Fri Feb 9 15:41:14 2018 by

LaTeXML
field of algebraic numbers

field of algebraic numbers

As special cases of the

theorem

of the parent
“

polynomial equation with algebraic coefficients

(

http://

planetmath

.org/polynomialequationwithalgebraiccoefficients

)” of this entry, one obtains the

Corollary.

If

α

and

β

are

algebraic numbers

, then also

α

+

β

,

α

-

β

,

α

⁢

β

and

α

β

(provided

β

≠

0

) are algebraic numbers.  If

α

and

β

are

algebraic integers

, then also

α

+

β

,

α

-

β

and

α

⁢

β

are algebraic integers.

The case of

α

β

needs an additional consideration:  If

x

m

+

b

1

⁢

x

m

-

1

+

…

+

b

m

-

1

⁢

x

+

b

m

is the

minimal polynomial

of

β

, the equation

β

m

+

b

1

⁢

β

m

-

1

+

…

+

b

m

-

1

⁢

β

+

b

m

=

0

implies

(

1

β

)

m

+

b

m

-

1

b

m

⁢

(

1

β

)

m

-

1

+

…

+

b

1

b

m

⋅

1

β

+

1

b

m

=

0

.

Hence

1

β

is an algebraic number, and therefore also

α

⋅

1

β

.

It follows from the corollary that the set of all algebraic numbers is a field and the set of all algebraic integers is a ring (an

integral domain

, too).  Moreover, the mentioned theorem implies that the

field of algebraic numbers

is

algebraically closed

and the

ring of algebraic integers

integrally closed

.  The field of algebraic numbers, which is sometimes denoted by

𝔸

, contains for example the

complex numbers

obtained from

rational numbers

by using

arithmetic operations

and taking

http://planetmath.org/node/5667

roots (these numbers form a

subfield

of

𝔸

).

Title

field of algebraic numbers

Canonical name

FieldOfAlgebraicNumbers

Date of creation

2015-11-18 14:30:41

Last modified on

2015-11-18 14:30:41

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

14

Author

pahio (2872)

Entry type

Definition

Classification

msc 11R04

Related topic

AlgebraicSumAndProduct

Related topic

SubfieldCriterion

Related topic

AlgebraicNumbersAreCountable

Related topic

RingWithoutIrreducibles

Related topic

AllAlgebraicNumbersInASequence

Defines

ring of algebraic integers

Generated on Fri Feb 9 15:41:20 2018 by

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Gaussian integer

Gaussian integer

A

complex number

of the form

a

+

b

⁢

i

, where

a

,

b

∈

ℤ

, is called
a

Gaussian integer

.

It is easy to see that the set

S

of all Gaussian integers is a

subring

of

ℂ

; specifically,

S

is the smallest subring containing

{

1

,

i

}

, whence

S

=

ℤ

⁢

[

i

]

.

ℤ

⁢

[

i

]

is a

Euclidean ring

, hence a

principal ring

, hence a

unique factorization domain

.

There are four units (i.e.

invertible elements

)
in the ring

ℤ

⁢

[

i

]

, namely

±

1

and

±

i

.
Up to

multiplication

by units, the primes in

ℤ

⁢

[

i

]

are

•

ordinary

prime numbers

≡

3

mod

4

•

elements of the form

a

±

b

⁢

i

where

a

2

+

b

2

is an ordinary
prime

≡

1

mod

4

(see Thue’s lemma)

•

the

element

1

+

i

.

Using the ring of Gaussian integers, it is not hard to show, for example,
that the

Diophantine equation

x

2

+

1

=

y

3

has no solutions

(

x

,

y

)

∈

ℤ

×

ℤ

except

(

0

,

1

)

.

Title

Gaussian integer

Canonical name

GaussianInteger

Date of creation

2013-03-22 11:45:32

Last modified on

2013-03-22 11:45:32

Owner

Daume (40)

Last modified by

Daume (40)

Numerical id

11

Author

Daume (40)

Entry type

Definition

Classification

msc 11R04

Classification

msc 55-00

Classification

msc 55U05

Classification

msc 32M10

Classification

msc 32C11

Classification

msc 14-02

Classification

msc 18-00

Related topic

EisensteinIntegers

Generated on Fri Feb 9 15:41:27 2018 by

LaTeXML
Gaussian prime

Gaussian prime

A

Gaussian prime

p

is a

Gaussian integer

a

+

b

⁢

i

(where

i

is the

imaginary unit

and

a

and

b

are real

integers

) that is

divisible

only by the units 1,

-

1

,

i

and

-

i

, itself, its

associates

and no others. For example,

3

+

20

⁢

i

is a Gaussian prime because there is no pair of Gaussian integers (besides the units and associates) that multiply to

3

+

20

⁢

i

. But

3

+

21

⁢

i

is not a Gaussian prime because

3

⁢

(

-

i

)

⁢

(

1

+

i

)

⁢

(

1

+

2

⁢

i

)

2

=

3

+

21

⁢

i

. If

a

+

b

⁢

i

is prime then so are

a

-

b

⁢

i

,

-

a

+

b

⁢

i

and

-

a

-

b

⁢

i

, as well as the associates

b

+

a

⁢

i

,

b

-

a

⁢

i

,

b

-

a

⁢

i

and

-

b

-

a

⁢

i

.

The real and the

imaginary parts

must be of different parity. For a

real prime

to be a Gaussian prime of the form

p

+

0

⁢

i

, the

real part

has to be of the form

p

=

4

⁢

n

-

1

; the same goes for the associates

0

+

p

⁢

i

. It follows from Fermat’s theorem on

sums of two squares

(

http://planetmath.org/RepresentingPrimesAsX2ny2

) that since real primes of the form

p

=

4

⁢

n

+

1

can be represented as

x

2

+

y

2

, then in the

complex plane

they have the factorization

(

x

+

y

⁢

i

)

⁢

(

x

-

y

⁢

i

)

. For example,

17

=

4

2

+

1

2

, so

(

4

+

i

)

⁢

(

4

-

i

)

=

17

.

Sometimes Gaussian primes are simply called “complex primes,” which is an incorrect term found in some of the older literature.

References

1

Kogbetliantz, Ervand George

Handbook of first complex prime numbers

London: Gordon and Breach Science Publishers (1971)

Title

Gaussian prime

Canonical name

GaussianPrime

Date of creation

2013-03-22 16:54:13

Last modified on

2013-03-22 16:54:13

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

7

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 11R04

Classification

msc 11A41

Generated on Fri Feb 9 15:41:29 2018 by

LaTeXML
ideal classes form an abelian group

ideal classes form an abelian group

Let

K

be a

number field

, and let

𝒞

be the set of

ideal classes

of

K

, with

multiplication

⋅

defined by

[

𝔞

]

⋅

[

𝔟

]

=

[

𝔞

⁢

𝔟

]

where

𝔞

,

𝔟

are ideals of

𝒪

K

.

We shall check the group properties:

1.

Associativity

:

[

𝔞

]

⋅

(

[

𝔟

]

⋅

[

𝔠

]

)

=

[

𝔞

]

⋅

[

𝔟

⁢

𝔠

]

=

[

𝔞

⁢

(

𝔟

⁢

𝔠

)

]

=

[

𝔞

⁢

𝔟

⁢

𝔠

]

=

[

(

𝔞

⁢

𝔟

)

⁢

𝔠

]

=

[

𝔞

⁢

𝔟

]

⋅

[

𝔠

]

=

(

[

𝔞

]

⋅

[

𝔟

]

)

⋅

[

𝔠

]

2.

Identity element

:

[

𝒪

K

]

⋅

[

𝔟

]

=

[

𝔟

]

=

[

𝔟

]

⋅

[

𝒪

K

]

.

3.

Inverses

: Consider

[

𝔟

]

. Let

b

be an

integer

in

𝔟

. Then

𝔟

⊇

(

b

)

, so there exists

𝔠

such that

𝔟

⁢

𝔠

=

(

b

)

.

Then the ideal class

[

𝔟

]

⋅

[

𝔠

]

=

[

(

b

)

]

=

[

𝒪

K

]

.

Then

𝒞

is a group under the

operation

⋅

.

It is

abelian

since

[

𝔞

]

⁢

[

𝔟

]

=

[

𝔞

⁢

𝔟

]

=

[

𝔟

⁢

𝔞

]

=

[

𝔟

]

⁢

[

𝔞

]

.

This is group is called the

ideal class group

of

K

. The ideal class group is one of the principal objects of

algebraic number theory

. In particular, for an arbitrary number field

K

, very little is known about the size of this group, called the

class number

of

K

. See the

analytic

class number formula

.

Title

ideal classes form an

abelian group

Canonical name

IdealClassesFormAnAbelianGroup

Date of creation

2013-03-22 12:49:40

Last modified on

2013-03-22 12:49:40

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

13

Author

mathcam (2727)

Entry type

Theorem

Classification

msc 11R04

Classification

msc 11R29

Related topic

NumberField

Related topic

ClassNumbersAndDiscriminantsTopicsOnClassGroups

Related topic

FractionalIdealOfCommutativeRing

Defines

ideal class group

Defines

class number

Generated on Fri Feb 9 15:41:32 2018 by

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ideal norm

ideal norm

Let

α

and

β

be

algebraic integers

in an

algebraic number field

K

and

𝔪

a non-zero ideal in the

ring of integers

of

K

.  We say that

α

and

β

are

congruent modulo the ideal

𝔪

in the case that

α

-

β

∈

𝔪

.  This is denoted by

α

≡

β

(

mod

𝔪

)

.

This

congruence relation

the ring of integers of

K

into

equivalence classes

, which are called the

residue classes

modulo the ideal

𝔪

.

Definition.

Let

K

be an algebraic number field and

𝔞

a non-zero ideal in

K

.  The

absolute norm of ideal

𝔞

means the number of all residue classes modulo

𝔞

.

Remark.

The of any ideal

𝔞

of

K

is finite — it has the expression

N

⁢

(

𝔞

)

=

Δ

⁢

(

𝔞

)

d

where

Δ

⁢

(

𝔞

)

is the

discriminant

of the ideal and

d

the

fundamental number

of the field.

•

N

⁢

(

𝔞

⁢

𝔟

)

=

N

⁢

(

𝔞

)

⋅

N

⁢

(

𝔟

)

•

N

⁢

(

𝔞

)

=

1

⇔

𝔞

=

(

1

)

•

N

⁢

(

(

α

)

)

=

|

N

⁢

(

α

)

|

•

N

⁢

(

𝔞

)

∈

𝔞

•

If

N

⁢

(

𝔭

)

is a

rational prime

, then

𝔭

is a

prime ideal

.

Title

ideal norm

Canonical name

IdealNorm

Date of creation

2013-03-22 15:43:23

Last modified on

2013-03-22 15:43:23

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

17

Author

pahio (2872)

Entry type

Definition

Classification

msc 11R04

Synonym

norm of an ideal

Synonym

norm of ideal

Related topic

NormAndTraceOfAlgebraicNumber

Related topic

Congruences

Related topic

MultiplicativeCongruence

Related topic

BasisOfIdealInAlgebraicNumberField

Related topic

IdealClassGroupIsFinite

Related topic

RationalIntegersInIdeals

Defines

congruence modulo an ideal

Defines

congruent modulo the ideal

Defines

residue classes modulo the ideal

Defines

absolute norm of ideal

Generated on Fri Feb 9 15:41:34 2018 by

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independence of characteristic polynomial on primitive element

independence of characteristic polynomial on primitive element

The

simple field extension

ℚ

⁢

(

ϑ

)

/

ℚ

where

ϑ

is an

algebraic number

of degree (

http://planetmath.org/DegreeOfAnAlgebraicNumber

)

n

may be determined also by using another

primitive element

η

.  Then we have

η

∈

ℚ

⁢

(

ϑ

)

,

whence, by the entry

degree of algebraic number

, the degree of

η

divides the degree of

ϑ

.  But also

ϑ

∈

ℚ

⁢

(

η

)

,

whence the degree of

ϑ

divides the degree of

η

.  Therefore any possible primitive element of the

field extension

has the same degree

n

.  This number is the degree of the

number field

(

http://planetmath.org/NumberField

), i.e. the degree of the field extension, as comes clear from the entry

canonical form of element of number field

.

Although the

characteristic polynomial

g

⁢

(

x

)

:=

∏

i

=

1

n

[

x

-

r

⁢

(

ϑ

i

)

]

=

∏

i

=

1

n

(

x

-

α

(

i

)

)

of an element

α

of the

algebraic number field

ℚ

⁢

(

ϑ

)

is based on the primitive element

ϑ

, the equation

g

⁢

(

x

)

=

(

x

-

α

1

)

m

⁢

(

x

-

α

2

)

m

⁢

⋯

⁢

(

x

-

α

k

)

m

(1)

in the entry

http://planetmath.org/node/12050

degree of algebraic number shows that the

polynomial

is fully determined by the

algebraic conjugates

of

α

itself and the number

m

which equals the degree

n

divided by the degree

k

of

α

.

The above stated makes meaningful to define the norm and the trace functions in an algebraic number field as follows.

Definition.

If

α

is an element of the number field

ℚ

⁢

(

ϑ

)

, then the

norm

N

⁢

(

α

)

and the

trace

S

⁢

(

α

)

of

α

are the

product

and the sum, respectively, of all

http://planetmath.org/node/12046

ℚ

⁢

(

ϑ

)

-

conjugates

α

(

i

)

of

α

.

Since the coefficients of the

characteristic equation

of

α

are

rational

, one has

N

:

ℚ

⁢

(

ϑ

)

→

ℚ

and

S

:

ℚ

⁢

(

ϑ

)

→

ℚ

.

In fact, one can infer from (1) that

N

⁢

(

α

)

=

a

k

m

,

S

⁢

(

α

)

=

-

m

⁢

a

1

,

(2)

where

x

k

+

a

1

⁢

x

k

-

1

+

…

+

a

k

is the

minimal polynomial

of

α

.

Title

independence of characteristic polynomial on primitive element

Canonical name

IndependenceOfCharacteristicPolynomialOnPrimitiveElement

Date of creation

2014-02-04 8:07:18

Last modified on

2014-02-04 8:07:18

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

10

Author

pahio (2872)

Entry type

Topic

Classification

msc 11R04

Classification

msc 12F05

Classification

msc 11C08

Classification

msc 12E05

Synonym

norm and trace functions in number field

Related topic

Norm

Related topic

NormAndTraceOfAlgebraicNumber

Related topic

PropertiesOfMathbbQvarthetaConjugates

Related topic

DiscriminantInAlgebraicNumberField

Defines

norm in number field

Defines

trace in number field

Defines

norm

Defines

trace

Generated on Fri Feb 9 15:41:36 2018 by

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integral basis

integral basis

Let

K

be a

number field

. A set of

algebraic integers

{

α

1

,

…

,

α

s

}

is said to be an

integral basis

for

K

if every

γ

in

𝒪

K

can be represented uniquely
as an

integer

linear combination

of

{

α

1

,

…

,

α

s

}

(i.e. one can write

γ

=

m

1

⁢

α

1

+

⋯

+

m

s

⁢

α

s

with

m

1

,

…

,

m

s

(

rational

) integers).

If

I

is an ideal of

𝒪

K

, then

{

α

1

,

…

,

α

s

}

∈

I

is said to be an

integral basis

for

I

if every element of

I

can be represented uniquely as an integer linear combination of

{

α

1

,

…

,

α

s

}

.

(In the above,

𝒪

K

denotes the

ring of algebraic integers

of

K

.)

An integral basis for

K

over

ℚ

is a basis for

K

over

ℚ

.

Title

integral basis

Canonical name

IntegralBasis

Date of creation

2013-03-22 12:36:03

Last modified on

2013-03-22 12:36:03

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

12

Author

rspuzio (6075)

Entry type

Definition

Classification

msc 11R04

Synonym

minimal basis

Synonym

minimal bases

Related topic

AlgebraicInteger

Related topic

Integral

Related topic

Basis

Related topic

DiscriminantOfANumberField

Related topic

ConditionForPowerBasis

Related topic

BasisOfIdealInAlgebraicNumberField

Related topic

CanonicalFormOfElementOfNumberField

Defines

integral bases

Generated on Fri Feb 9 15:41:38 2018 by

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