aarjaneiro/planetmath huggingface dataset:
Lindemann-Weierstrass theorem

Lindemann-Weierstrass theorem

If

α

1

,

…

,

α

n

are

linearly independent

algebraic numbers

over

ℚ

, then

e

α

1

,

…

,

e

α

n

are

algebraically independent

over

ℚ

.

An

equivalent

version of the

theorem

that if

α

1

,

…

,

α

n

are distinct algebraic numbers over

ℚ

, then

e

α

1

,

…

,

e

α

n

are linearly independent over

ℚ

.

Some immediate

consequences

of this theorem:

•

If

α

is a non-zero algebraic number over

ℚ

, then

e

α

is transcendental over

ℚ

.

•

e

is transcendental over

ℚ

.

•

π

is transcendental over

ℚ

. As a result, it is impossible to “square the circle”!

It is easy to see that

π

is transcendental over

ℚ

⁢

(

e

)

iff

e

is transcendental over

ℚ

⁢

(

π

)

iff

π

and

e

are algebraically independent. However, whether

π

and

e

are algebraically independent is still an

open question

today.

Schanuel’s conjecture is a

generalization

of the

Lindemann-Weierstrass theorem

. If Schanuel’s conjecture were proven to be true, then the algebraic independence of

e

and

π

over

ℚ

can be shown.

Title

Lindemann-Weierstrass theorem

Canonical

name

LindemannWeierstrassTheorem

Date of creation

2013-03-22 14:19:22

Last modified on

2013-03-22 14:19:22

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

11

Author

CWoo (3771)

Entry type

Theorem

Classification

msc 12D99

Classification

msc 11J85

Synonym

Lindemann’s theorem

Related topic

SchanuelsConjecutre

Related topic

GelfondsTheorem

Related topic

Irrational

Related topic

EIsTranscendental

Generated on Fri Feb 9 16:39:49 2018 by

LaTeXML
proof of the fundamental theorem of algebra (Liouville’s theorem)

proof of the fundamental theorem of algebra (Liouville’s theorem)

Let

f

:

ℂ

→

ℂ

be a

polynomial

, and suppose

f

has no root in

ℂ

. We will show

f

is constant.

Let

g

=

1

f

. Since

f

is never zero,

g

is defined and

holomorphic

on

ℂ

(ie. it is entire). Moreover, since

f

is a polynomial,

|

f

⁢

(

z

)

|

→

∞

as

|

z

|

→

∞

, and so

|

g

⁢

(

z

)

|

→

0

as

|

z

|

→

∞

. Then there is some

M

>

0

such that

|

g

⁢

(

z

)

|

<

1

whenever

|

z

|

>

M

, and

g

is

continuous

and so

bounded

on the

compact set

{

z

∈

ℂ

:

|

z

|

≤

M

}

.

So

g

is bounded and entire, and therefore by Liouville’s theorem

g

is constant. So

f

is constant as required.

□

Title

proof of the

fundamental theorem of algebra

(Liouville’s theorem)

Canonical name

ProofOfTheFundamentalTheoremOfAlgebraLiouvillesTheorem

Date of creation

2013-03-22 12:18:59

Last modified on

2013-03-22 12:18:59

Owner

Evandar (27)

Last modified by

Evandar (27)

Numerical id

6

Author

Evandar (27)

Entry type

Proof

Classification

msc 12D99

Classification

msc 30A99

Generated on Fri Feb 9 16:39:54 2018 by

LaTeXML
real and complex embeddings

real and complex embeddings

Let

L

be a

subfield

of

ℂ

.

Definition 1

.

1.

A

real embedding

of

L

is an injective

field
homomorphism

σ

:

L

↪

ℝ

2.

A (non-real)

complex embedding

of

L

is an injective
field homomorphism

τ

:

L

↪

ℂ

such that

τ

⁢

(

L

)

⊈

ℝ

.

3.

We denote

Σ

L

the set of all

embeddings

, real and

complex

, of

L

in

ℂ

(note that all of them must fix

ℚ

, since they are field homomorphisms).

Note that if

σ

is a real embedding then

σ

¯

=

σ

, where

⋅

¯

denotes the

complex conjugation

automorphism

:

⋅

¯

:

ℂ

→

ℂ

,

(

a

+

b

⁢

i

)

¯

=

a

-

b

⁢

i

On the other
hand, if

τ

is a complex embedding, then

τ

¯

is
another complex embedding, so the complex embeddings always come
in pairs

{

τ

,

τ

¯

}

.

Let

K

⊆

L

be another subfield of

ℂ

. Moreover,
assume that

[

L

:

K

]

is finite (this is the

dimension

of

L

as a

vector space

over

K

). We are interested in the embeddings of

L

that fix

K

pointwise

, i.e. embeddings

ψ

:

L

↪

ℂ

such that

ψ

⁢

(

k

)

=

k

,

∀

k

∈

K

Theorem 1

.

For any embedding

ψ

of

K

in

C

, there are exactly

[

L

:

K

]

embeddings of

L

such that they extend

ψ

. In other
words, if

φ

is one of them, then

φ

⁢

(

k

)

=

ψ

⁢

(

k

)

,

∀

k

∈

K

Thus, by taking

ψ

=

Id

K

, there are exactly

[

L

:

K

]

embeddings of

L

which fix

K

pointwise.

Hence, by the theorem, we know that the order of

Σ

L

is

[

L

:

ℚ

]

. The number

[

L

:

ℚ

]

is usually decomposed as

[

L

:

ℚ

]

=

r

1

+

2

r

2

where

r

1

is the number of embeddings which are real, and

2

⁢

r

2

is the number of embeddings which are complex (non-real). Notice
that by the remark above this number is always even, so

r

2

is
an integer.

Remark

: Let

ψ

be an embedding of

L

in

ℂ

.
Since

ψ

is injective, we have

ψ

⁢

(

L

)

≅

L

, so we can
regard

ψ

as an automorphism of

L

. When

L

/

ℚ

is a

Galois extension

, we can prove that

Σ

L

≅

Gal

⁡

(

L

/

ℚ

)

, and hence proving in a different way
the fact that

∣

Σ

L

∣

=

[

L

:

ℚ

]

=

∣

Gal

(

L

/

ℚ

)

∣

Title

real and complex embeddings

Canonical name

RealAndComplexEmbeddings

Date of creation

2013-03-22 13:54:43

Last modified on

2013-03-22 13:54:43

Owner

alozano (2414)

Last modified by

alozano (2414)

Numerical id

4

Author

alozano (2414)

Entry type

Definition

Classification

msc 12D99

Related topic

GaloisGroup

Related topic

TotallyRealAndImaginaryFields

Related topic

RamificationOfArchimedeanPlaces

Defines

real embedding

Defines

complex embedding

Generated on Fri Feb 9 16:39:56 2018 by

LaTeXML
some values characterising i

some values characterising i

•

i

2

=

-

1

•

|

i

|

=

1

•

arg

⁡

i

=

2

⁢

n

⁢

π

+

π

2

(

n

∈

ℤ

)

•

i

¯

=

-

i

•

1

i

=

-

i

•

i

=

±

1

+

i

2

•

(

-

1

)

i

=

e

(

2

⁢

n

+

1

)

⁢

π

(

n

∈

ℤ

)

•

i

i

=

e

2

⁢

n

⁢

π

-

π

2

(

n

∈

ℤ

)

•

cos

⁡

i

=

1

2

⁢

(

e

+

1

e

)

≈

1.54308

•

sin

⁡

i

=

i

2

⁢

(

e

-

1

e

)

≈

1.17520

⁢

i

•

cosh

⁡

i

=

cos

⁡

1

≈

0.54030

•

sinh

⁡

i

=

i

⁢

sin

⁡

1

≈

0.84147

⁢

i

•

e

i

=

cos

⁡

1

+

i

⁢

sin

⁡

1

•

log

⁡

i

=

(

2

⁢

n

⁢

π

+

π

2

)

⁢

i

(

n

∈

ℤ

)

Title

some values characterising i

Canonical name

SomeValuesCharacterisingI

Date of creation

2013-03-22 18:31:29

Last modified on

2013-03-22 18:31:29

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

7

Author

pahio (2872)

Entry type

Result

Classification

msc 12D99

Related topic

GeneralPower

Related topic

ComplexSineAndCosine

Related topic

ComplexLogarithm

Generated on Fri Feb 9 16:40:06 2018 by

LaTeXML
subinterval

subinterval

Let

I

be an interval. A

subinterval

of

I

is a subset of

I

that is also an interval.

Title

subinterval

Canonical

name

Subinterval

Date of creation

2013-03-22 17:51:27

Last modified on

2013-03-22 17:51:27

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

4

Author

Wkbj79 (1863)

Entry type

Definition

Classification

msc 12D99

Classification

msc 26-00

Classification

msc 54C30

Related topic

Partition3

Generated on Fri Feb 9 16:40:08 2018 by

LaTeXML
totally real and imaginary fields

totally real and imaginary fields

For this entry, we follow the notation of the entry

real and
complex embeddings

.

Let

K

be a

subfield

of the

complex numbers

,

ℂ

, and let

Σ

K

be the set of all embeddings of

K

in

ℂ

.

Definition 1

.

With

K

as above:

1.

K

is a

totally real field

if all embeddings

ψ

∈

Σ

K

are real embeddings.

2.

K

is a

totally imaginary field

if all embeddings

ψ

∈

Σ

K

are (non-real) complex embeddings.

3.

K

is a

CM-field

or

complex multiplication

field

if

K

is a totally imaginary

quadratic extension

of a totally real
field.

Note that, for example, one can obtain a CM-field

K

from a totally real number field

F

by adjoining the

square root

of a number all of whose

conjugates

are

negative

.

Note: A complex number

ω

is real if and only if

ω

¯

, the

complex conjugate

of

ω

, equals

ω

:

ω

∈

ℝ

⇔

ω

=

ω

¯

Thus, a field

K

which is fixed

pointwise

by complex
conjugation is real (i.e.

strictly

contained in

ℝ

). However,

K

might not be

totally real

. For example, let

α

be the unique real third root of

2

. Then

ℚ

⁢

(

α

)

is real but not totally real.

Given a field

L

, the subfield of

L

fixed pointwise by complex conjugation is called the

maximal real subfield of

L

.

For examples (of

(

1

)

,

(

2

)

and

(

3

)

), see

examples of totally real fields

.

Title

totally real and

imaginary

fields

Canonical

name

TotallyRealAndImaginaryFields

Date of creation

2013-03-22 13:55:02

Last modified on

2013-03-22 13:55:02

Owner

alozano (2414)

Last modified by

alozano (2414)

Numerical id

8

Author

alozano (2414)

Entry type

Definition

Classification

msc 12D99

Synonym

complex multiplication field

Related topic

RealAndComplexEmbeddings

Related topic

TotallyImaginaryExamplesOfTotallyReal

Related topic

ExamplesOfRamificationOfArchimedeanPlaces

Defines

totally real field

Defines

totally imaginary field

Defines

CM-field

Defines

maximal real subfield

Generated on Fri Feb 9 16:40:10 2018 by

LaTeXML
degree of an algebraic number

degree of an algebraic number

Let

α

be an

algebraic number

. The

degree

of

α

is the degree (

http://planetmath.org/Degree8

) of the

minimal polynomial

for

α

over

ℚ

.

In a manner to

polynomials

, the degree of

α

may be denoted

deg

⁡

α

.

For example, since

x

3

-

2

is the minimal polynomial for

2

3

over

ℚ

, we have

deg

⁡

2

3

=

3

.

Title

degree of an algebraic number

Canonical name

DegreeOfAnAlgebraicNumber

Date of creation

2013-03-22 17:50:05

Last modified on

2013-03-22 17:50:05

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

4

Author

Wkbj79 (1863)

Entry type

Definition

Classification

msc 12E05

Classification

msc 12F05

Classification

msc 11C08

Classification

msc 11R04

Related topic

AlgebraicNumber

Related topic

Degree8

Related topic

MinimalPolynomial

Related topic

TheoryOfAlgebraicNumbers

Generated on Fri Feb 9 16:40:12 2018 by

LaTeXML
Gauss’s lemma I

Gauss’s lemma I

There are a few different things that are sometimes called “Gauss’s Lemma”. See also Gauss’s Lemma II.

Gauss’s Lemma I:

If

R

is a UFD and

f

⁢

(

x

)

and

g

⁢

(

x

)

are both

primitive polynomials

in

R

⁢

[

x

]

, so is

f

⁢

(

x

)

⁢

g

⁢

(

x

)

.

Proof:

Suppose

f

⁢

(

x

)

⁢

g

⁢

(

x

)

not

primitive

. We will show either

f

⁢

(

x

)

or

g

⁢

(

x

)

isn’t as well.

f

⁢

(

x

)

⁢

g

⁢

(

x

)

not primitive means that there exists some non-unit

d

in

R

that divides all the

coefficients

of

f

⁢

(

x

)

⁢

g

⁢

(

x

)

. Let

p

be an

irreducible

factor of

d

, which exists and is a

prime element

because

R

is a UFD. We consider the

quotient ring

of

R

by the

principal ideal

p

⁢

R

generated by

p

, which is a

prime ideal

since

p

is a prime element. The

canonical projection

R

→

R

/

p

⁢

R

induces a surjective

ring homomorphism

θ

:

R

⁢

[

X

]

→

(

R

/

p

⁢

R

)

⁢

[

X

]

, whose kernel consists of all

polynomials

all of whose coefficients are divisible by

p

; these polynomials are therefore not primitive.

Since

p

⁢

R

is a prime ideal,

R

/

p

⁢

R

is an

integral domain

, so

(

R

/

p

⁢

R

)

⁢

[

x

]

is also an integral domain. By hypothesis

θ

sends the

product

f

⁢

(

x

)

⁢

g

⁢

(

x

)

to

0

∈

(

R

/

p

⁢

R

)

⁢

[

X

]

, which is therefore the product of

θ

⁢

(

f

⁢

(

x

)

)

and

θ

⁢

(

g

⁢

(

x

)

)

, and one of these two factors in

(

R

/

p

⁢

R

)

⁢

[

x

]

must be zero. But that means that

f

⁢

(

x

)

or

g

⁢

(

x

)

is in the kernel of

θ

, and therefore not primitive.

Title

Gauss’s lemma I

Canonical name

GausssLemmaI

Date of creation

2013-03-22 13:07:49

Last modified on

2013-03-22 13:07:49

Owner

bshanks (153)

Last modified by

bshanks (153)

Numerical id

17

Author

bshanks (153)

Entry type

Theorem

Classification

msc 12E05

Related topic

GausssLemmaII

Generated on Fri Feb 9 16:40:21 2018 by

LaTeXML
Gauss’s lemma II

Gauss’s lemma II

Definition.

A

polynomial

P

=

a

n

⁢

x

n

+

⋯

+

a

0

over an

integral domain

D

is said to be

primitive

if its

coefficients

are not all

divisible

by any element of

D

other than a unit.

Proposition

(Gauss).

Let

D

be a

unique factorization domain

and

F

its

field of fractions

.
If a polynomial

P

∈

D

⁢

[

x

]

is reducible in

F

⁢

[

x

]

, then it is reducible in

D

⁢

[

x

]

.

Remark.

The above statement is often used in its

contrapositive

form.  For an example of this usage, see this entry (

http://planetmath.org/AlternativeProofThatSqrt2IsIrrational

).

Proof.

A

primitive polynomial

in

D

⁢

[

x

]

is by definition divisible by a non

invertible

constant polynomial

, and therefore reducible in

D

⁢

[

x

]

(unless it is itself

constant

). There is therefore nothing to prove unless

P

(which is not constant) is primitive.  By

assumption

there exist non-constant

S

,

T

∈

F

⁢

[

x

]

such that

P

=

S

⁢

T

.  There are

elements

a

,

b

∈

F

such that

a

⁢

S

and

b

⁢

T

are in

D

⁢

[

x

]

and are primitive (first multiply by a nonzero element of

D

to chase any

denominators

, then divide by the gcd of the resulting coefficients in

D

).  Then

a

⁢

S

⁢

b

⁢

T

=

a

⁢

b

⁢

P

is primitive by Gauss’s lemma I, but

P

is primitive as well, so

a

⁢

b

is a unit of

D

and

P

=

(

a

⁢

b

)

-

1

⁢

(

a

⁢

S

)

⁢

(

b

⁢

T

)

is a nontrivial decomposition of

P

in

D

⁢

[

X

]

.  This

completes

the proof.

Remark.

Another result with the same name is Gauss’ lemma on

quadratic residues

.

From the above proposition and its proof one may infer the

Theorem

.

If a primitive polynomial of

D

⁢

[

x

]

is divisible in

F

⁢

[

x

]

, then it splits in

D

⁢

[

x

]

into

primitive prime factors

.  These are uniquely determined up to unit factors of

D

.

Title

Gauss’s lemma II

Canonical name

GausssLemmaII

Date of creation

2013-03-22 13:07:52

Last modified on

2013-03-22 13:07:52

Owner

bshanks (153)

Last modified by

bshanks (153)

Numerical id

18

Author

bshanks (153)

Entry type

Theorem

Classification

msc 12E05

Synonym

Gauss’ lemma II

Related topic

GausssLemmaI

Related topic

EisensteinCriterion

Related topic

ProofOfEisensteinCriterion

Related topic

PrimeFactorsOfXn1

Related topic

AlternativeProofThatSqrt2IsIrrational

Defines

primitive polynomial

Generated on Fri Feb 9 16:40:22 2018 by

LaTeXML
polynomial analogon for Fermat’s last theorem

polynomial analogon for Fermat’s last theorem

For

polynomials

with

complex

coefficients, there is an analogon of Fermat’s last

theorem

.  It can be proven quite elementarily by using Mason’s theorem (1983), but the original proof (about in 1900) was based on methods of

algebraic geometry

.

Theorem.

For an

integer

n

greater than 2, there exist no non-constant

coprime

polynomials

x

⁢

(

t

)

,

y

⁢

(

t

)

,

z

⁢

(

t

)

in the ring

ℂ

⁢

[

t

]

satisfying

[

x

⁢

(

t

)

]

n

+

[

y

⁢

(

t

)

]

n

=

[

z

⁢

(

t

)

]

n

.

(1)

Remark.

For

n

=

2

, the equation (1) is in e.g. as

(

2

⁢

t

)

2

+

(

1

-

t

2

)

2

=

(

1

+

t

2

)

2

.

References

1

Serge Lang

: “Die

a

⁢

b

⁢

c

-Vermutung”.  –

Elemente der Mathematik

48

(1993).

Title

polynomial analogon for Fermat’s last theorem

Canonical

name

PolynomialAnalogonForFermatsLastTheorem

Date of creation

2013-03-22 19:13:05

Last modified on

2013-03-22 19:13:05

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

8

Author

pahio (2872)

Entry type

Theorem

Classification

msc 12E05

Classification

msc 11C08

Synonym

Fermat’s last theorem for polynomials

Related topic

MasonsTheorem

Related topic

WeierstrassSubstitutionFormulas

Generated on Fri Feb 9 16:40:28 2018 by

LaTeXML
translation automorphism of a polynomial ring

translation automorphism of a polynomial ring

Let

R

be a

commutative ring

, let

R

⁢

[

X

]

be the

polynomial ring

over

R

, and let

a

be an element of

R

. Then we can define a

homomorphism

τ

a

of

R

⁢

[

X

]

by constructing the

evaluation homomorphism

from

R

⁢

[

X

]

to

R

⁢

[

X

]

taking

r

∈

R

to itself and taking

X

to

X

+

a

.

To see that

τ

a

is an

automorphism

, observe that

τ

-

a

∘

τ

a

is the

identity

on

R

⊂

R

⁢

[

X

]

and takes

X

to

X

, so by the uniqueness of the evaluation homomorphism,

τ

-

a

∘

τ

a

is the identity.

Title

translation

automorphism of a polynomial ring

Canonical name

TranslationAutomorphismOfAPolynomialRing

Date of creation

2013-03-22 14:16:13

Last modified on

2013-03-22 14:16:13

Owner

archibal (4430)

Last modified by

archibal (4430)

Numerical id

4

Author

archibal (4430)

Entry type

Example

Classification

msc 12E05

Classification

msc 11C08

Classification

msc 13P05

Related topic

IsomorphismSwappingZeroAndUnity

Generated on Fri Feb 9 16:40:38 2018 by

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Tschirnhaus transformations

Tschirnhaus transformations

A polynomial transformation which transforms a

polynomial

to another with certain zero-coefficients is called a

Tschirnhaus Transformation

. It is thus an

invertible

transformation

of the form

x

↦

g

⁢

(

x

)

/

h

⁢

(

x

)

where

g

,

h

are polynomials over the

base field

K

(or some

subfield

of the

splitting field

of the polynomial being transformed). If

gcd

⁡

(

h

⁢

(

x

)

,

f

⁢

(

x

)

)

=

1

then the Tschirnhaus transformation becomes a polynomial transformation mod f.

Specifically, it concerns a substitution that reduces finding the roots of the polynomial

p

=

T

n

+

a

1

⁢

T

n

-

1

+

…

+

a

n

=

∏

i

=

1

n

(

T

-

r

i

)

∈

k

⁢

[

T

]

to finding the roots of another q - with less

parameters

- and solving an auxiliary

polynomial equation

s, with

deg

⁡

(

s

)

<

deg

⁡

(

p

∩

q

)

.

Historically, the transformation was applied to reduce the general

quintic equation

, to simpler resolvents. Examples due to Hermite and Klein are
respectively: The principal resolvent

K

⁢

(

X

)

:=

X

5

+

a

0

⁢

X

2

+

a

1

⁢

X

+

a

3

and the Bring-Jerrard form

K

⁢

(

X

)

:=

X

5

+

a

1

⁢

X

+

a

2

Tschirnhaus transformations are also used when computing

Galois
groups

to remove repeated roots in resolvent polynomials. Almost any transformation will work but it is
extremely hard to find an efficient algorithm that can be proved
to work.

Title

Tschirnhaus transformations

Canonical

name

TschirnhausTransformations

Date of creation

2013-03-22 13:50:12

Last modified on

2013-03-22 13:50:12

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

11

Author

mathcam (2727)

Entry type

Definition

Classification

msc 12E05

Synonym

Tschirnhausen Transformation

Related topic

QuadraticResolvent

Related topic

EulersDerivationOfTheQuarticFormula

Generated on Fri Feb 9 16:40:40 2018 by

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all one polynomial

all one polynomial

An

all one

polynomial

(

AOP

) is a polynomial used in

finite fields

, specifically GF(

2

). The AOP is a 1-equally spaced polynomial.

An AOP of degree

m

can be written as follows:

AOP

⁡

(

x

)

=

∑

i

=

0

m

x

i

=

x

m

+

x

m

-

1

+

…

+

x

+

1

Over GF(

2

) the AOP has many interesting properties, including:

•

The

Hamming weight

of the AOP is

m

+

1

.

•

The AOP is

irreducible polynomial

iff

m

+

1

is prime and

2

is a

primitive root

modulo

m

+

1

.

•

The only AOP that is a

primitive polynomial

is

x

2

+

x

+

1

.

Despite the fact that the Hamming weight is large, because of the ease of representation and other improvements there are efficient hardware and software implementations for use in areas such as

coding theory

and

cryptography

.

Title

all one polynomial

Canonical name

AllOnePolynomial

Date of creation

2013-03-22 15:00:26

Last modified on

2013-03-22 15:00:26

Owner

Derk (34)

Last modified by

Derk (34)

Numerical id

7

Author

Derk (34)

Entry type

Definition

Classification

msc 12E10

Synonym

all-one polynomial

Synonym

AOP

Related topic

CyclotomicPolynomial

Related topic

ProofThatTheCyclotomicPolynomialIsIrreducible

Related topic

FactoringAllOnePolynomialsUsingTheGroupingMethod

Generated on Fri Feb 9 16:40:41 2018 by

LaTeXML
de Moivre identity

de Moivre identity

From the

Euler relation

e

i

⁢

θ

=

cos

⁡

θ

+

i

⁢

sin

⁡

θ

it follows that

e

i

⁢

θ

⋅

n

=

(

e

i

⁢

θ

)

n

cos

⁡

n

⁢

θ

+

i

⁢

sin

⁡

n

⁢

θ

=

(

cos

⁡

θ

+

i

⁢

sin

⁡

θ

)

n

where

n

∈

ℤ

.
This is called

de Moivre’s formula

, and besides being generally useful, it’s a convenient way to remember double- (and higher-multiple-) angle formulas. For example,

cos

⁡

2

⁢

θ

+

i

⁢

sin

⁡

2

⁢

θ

=

(

cos

⁡

θ

+

i

⁢

sin

⁡

θ

)

2

=

cos

2

⁡

θ

+

2

⁢

i

⁢

sin

⁡

θ

⁢

cos

⁡

θ

-

sin

2

⁡

θ

.

Since the

imaginary parts

and

real parts

on each side must be equal, we must have

cos

⁡

2

⁢

θ

=

cos

2

⁡

θ

-

sin

2

⁡

θ

and

sin

⁡

2

⁢

θ

=

2

⁢

sin

⁡

θ

⁢

cos

⁡

θ

.

Title

de Moivre identity

Canonical name

DeMoivreIdentity

Date of creation

2013-03-22 12:20:45

Last modified on

2013-03-22 12:20:45

Owner

Daume (40)

Last modified by

Daume (40)

Numerical id

11

Author

Daume (40)

Entry type

Theorem

Classification

msc 12E10

Synonym

de Moivre’s theorem

Synonym

de Moivre’s formula

Related topic

EulerRelation

Related topic

DoubleAngleIdentity

Related topic

ArgumentOfProductAndSum

Related topic

ArgumentOfProductAndQuotient

Generated on Fri Feb 9 16:40:43 2018 by

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