aarjaneiro/planetmath huggingface dataset:
two-generator property

two-generator property

Theorem.

Every ideal of a

Dedekind domain

can be generated by two of its elements.

Proof

.  Let

𝔞

be an arbitrary ideal of a Dedekind domain

R

.  Let

𝔟

be such an ideal of

R

that

𝔞

⁢

𝔟

is a

principal ideal

(

β

)

.  The lemma to which this entry is attached gives also an element

γ

and an ideal

𝔠

of

R

such that

𝔞

⁢

𝔠

=

(

γ

)

and

𝔟

+

𝔠

=

R

.  Then we have

𝔞

=

gcd

⁡

(

𝔞

⁢

𝔟

,

𝔞

⁢

𝔠

)

=

gcd

⁡

(

(

β

)

,

(

γ

)

)

=

(

β

,

γ

)

because

gcd

⁡

(

𝔟

,

𝔠

)

=

𝔟

+

𝔠

=

R

=

(

1

)

.

□

The Dedekind domains are trivially Prüfer domains, but the

two-generator property

can not be generalized to the

invertible ideals

of all Prüfer domains (and Prüfer rings):  Schülting has constructed an invertible ideal of a Prüfer domain that can not be generated by less than three

generators

.  The example of Schülting is the

fractional ideal

(

1

,

X

,

Y

)

of the Prüfer domain

⋂

j

B

j

where the

B

j

’s run all

valuation rings

of the

rational function field

ℝ

⁢

(

X

,

Y

)

which have the

residue fields

formally real

.

References

1

Eben Matlis:

“The two-generator problem for ideals”.  –

The Michigan Mathematical Journal

17

N

∘

3 (1970).

2

Heinz-Werner Schülting:

“Über die Erzeugendenanzahl invertierbarer Ideale in Prüferringen”.  –

Communications in

Algebra

7

N

∘

13 (1979). [Zentralblatt 432.13010]

Title

two-generator property

Canonical name

TwogeneratorProperty

Date of creation

2015-05-05 15:25:37

Last modified on

2015-05-05 15:25:37

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

38

Author

pahio (2872)

Entry type

Theorem

Classification

msc 11R04

Synonym

Ideal of Dedekind domain

Related topic

SumOfIdeals

Related topic

FamousAndInfamousOpenQuestionsInMathematics

Related topic

AnyDivisorIsGcdOfTwoPrincipalDivisors

Generated on Fri Feb 9 15:42:22 2018 by

LaTeXML
units of quadratic fields

units of quadratic fields

Dirichlet’s unit theorem gives all units of an

algebraic number field

ℚ

⁢

(

ϑ

)

in the unique form

ε

=

ζ

n

⁢

η

1

k

1

⁢

η

2

k

2

⁢

…

⁢

η

t

k

t

,

where

ζ

is a primitive

w

th

root of unity

in

ℚ

⁢

(

ϑ

)

, the

η

j

’s are the

fundamental units

of

ℚ

⁢

(

ϑ

)

,

0

≦

n

≦

w

-

1

,

k

j

∈

ℤ

∀

j

,

t

=

r

+

s

-

1

.

•

The case of a

real quadratic field

ℚ

⁢

(

m

)

, the

square-free

m

>

1

:

r

=

2

,

s

=

0

,

t

=

r

+

s

-

1

=

1

.  So we obtain

ε

=

ζ

n

⁢

η

k

=

±

η

k

,

because

ζ

=

-

1

is the only real

primitive root of unity

(

w

=

2

).  Thus, every real quadratic field has infinitely many units and a unique fundamental unit

η

.

Examples:  If

m

=

3

,  then

η

=

2

+

3

;  if

m

=

421

,  then

η

=

444939

+

21685

⁢

421

2

.

•

The case of any

imaginary quadratic field

ℚ

⁢

(

ϑ

)

; here

ϑ

=

m

,  the square-free

m

<

0

:  The

conjugates

of

ϑ

are the

pure imaginary numbers

±

m

, hence

r

=

0

,

2

⁢

s

=

2

,

t

=

r

+

s

-

1

=

0

.  Thus we see that all units are

ε

=

ζ

n

.

1)

m

=

-

1

.  The field contains the primitive fourth root of unity, e.g.

i

, and therefore all units in the

field

ℚ

⁢

(

i

)

are

i

n

, where

n

=

0

,

1

,

2

,

3

.

2)

m

=

-

3

.  The field in question is a

cyclotomic field

(

http://planetmath.org/CyclotomicExtension

) containing the primitive third root of unity and also the primitive sixth root of unity, namely

ζ

=

cos

⁡

2

⁢

π

6

+

i

⁢

sin

⁡

2

⁢

π

6

;

hence all units are

ε

=

(

1

+

-

3

2

)

n

,  where

n

=

0

,

1

,

…

,

5

, or, equivalently,

ε

=

±

(

-

1

+

-

3

2

)

n

,  where

n

=

0

,

1

,

2

.

3)

m

=

-

2

,

m

<

-

3

.   The only roots of unity in the field are

±

1

; hence

ζ

=

-

1

,

w

=

2

,  and the units of the field are simply

(

-

1

)

n

, where

n

=

0

,

1

.

Title

units of quadratic fields

Canonical name

UnitsOfQuadraticFields

Date of creation

2013-03-22 14:16:30

Last modified on

2013-03-22 14:16:30

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

39

Author

pahio (2872)

Entry type

Application

Classification

msc 11R04

Classification

msc 11R27

Synonym

quadratic unit

Related topic

Unit

Related topic

NumberField

Related topic

ImaginaryQuadraticField

Related topic

SomethingRelatedToFundamentalUnits

Generated on Fri Feb 9 15:42:25 2018 by

LaTeXML
Salem number

Salem number

Salem number

is a real

algebraic integer

α

>

1

whose

algebraic conjugates

all lie in the unit disk

{

z

∈

ℂ

|

|

z

|

≤

1

}

with at least one on the unit circle

{

z

∈

ℂ

|

|

z

|

=

1

}

.

Powers of a Salem number

α

n

(

n

=

1

,

2

,

…

)

are everywhere dense modulo

1

, but are not

uniformly distributed

modulo

1

.

The smallest known Salem number is the largest positive root of

α

10

+

α

9

-

α

7

-

α

6

-

α

5

-

α

4

-

α

3

+

α

+

1

=

0

.

Title

Salem number

Canonical name

SalemNumber

Date of creation

2013-03-22 13:38:48

Last modified on

2013-03-22 13:38:48

Owner

bbukh (348)

Last modified by

bbukh (348)

Numerical id

6

Author

bbukh (348)

Entry type

Definition

Classification

msc 11R06

Classification

msc 11J71

Generated on Fri Feb 9 15:42:27 2018 by

LaTeXML
class numbers of imaginary quadratic fields

class numbers of imaginary quadratic fields

We tabulate the

ideal class

numbers (

http://planetmath.org/ClassNumber

)

h

of first

imaginary quadratic number fields

ℚ

⁢

(

d

)

.  The table all the nine cases where the

class number

is 1.

d

h

d

h

d

h

d

h

-

1

1

!

-

47

5

-

97

4

-

146

16

-

2

1

!

-

51

2

-

101

14

-

149

14

-

3

1

!

-

53

6

-

102

4

-

151

7

-

5

2

-

55

4

-

103

5

-

154

8

-

6

2

-

57

4

-

105

8

-

155

4

-

7

1

!

-

58

2

-

106

6

-

157

6

-

10

2

-

59

3

-

107

3

-

158

8

-

11

1

!

-

61

6

-

109

6

-

159

10

-

13

2

-

62

8

-

110

12

-

161

16

-

14

4

-

65

8

-

111

8

-

163

1

!

-

15

2

-

66

8

-

113

8

-

165

8

-

17

4

-

67

1

!

-

114

8

-

166

10

-

19

1

!

-

69

8

-

115

2

-

167

11

-

21

4

-

70

4

-

118

6

-

170

12

-

22

2

-

71

7

-

119

10

-

173

14

-

23

3

-

73

4

-

122

10

-

174

12

-

26

6

-

74

10

-

123

2

-

177

4

-

29

6

-

77

8

-

127

5

-

178

8

-

30

4

-

78

4

-

129

12

-

179

5

-

31

3

-

79

5

-

130

4

-

181

10

-

33

4

-

82

4

-

131

5

-

182

12

-

34

4

-

83

3

-

133

4

-

183

8

-

35

2

-

85

4

-

134

14

-

185

16

-

37

2

-

86

10

-

137

8

-

186

12

-

38

6

-

87

6

-

138

8

-

187

2

-

39

4

-

89

12

-

139

3

-

190

4

-

41

8

-

91

2

-

141

8

-

191

13

-

42

4

-

93

4

-

142

4

-

193

4

-

43

1

!

-

94

8

-

143

10

-

194

20

-

46

4

-

95

8

-

145

8

-

195

4

The class numbers of

ℚ

⁢

(

d

)

for the

squarefree

d

’s form Sloane’s

http://www.research.att.com/ njas/

sequences

/?q=A000924&sort=0&fmt=0&

language

=english&go=Search

sequence A000924.

References

1

S. Borewicz & I. Safarevic

:

Zahlentheorie

.  Birkhäuser Verlag. Basel und Stuttgart (1966).

Title

class numbers of imaginary quadratic fields

Canonical name

ClassNumbersOfImaginaryQuadraticFields

Date of creation

2013-03-22 18:31:20

Last modified on

2013-03-22 18:31:20

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

10

Author

pahio (2872)

Entry type

Data Structure

Classification

msc 11R11

Classification

msc 11R04

Related topic

LemmaForImaginaryQuadraticFields

Related topic

QuadraticImaginaryEuclideanNumberFields

Generated on Fri Feb 9 15:42:30 2018 by

LaTeXML
imaginary quadratic field

imaginary quadratic field

A

quadratic number field

ℚ

⁢

(

d

)

,  where

d

is a

negative

squarefree

integer

, is called a

imaginary quadratic field

.

The name may be thought to describe the fact that such a

quadratic field

“

imaginary numbers

” (in the sense ‘non-real

complex numbers

’).

Other quadratic fields are

real quadratic fields

.

Title

imaginary quadratic field

Canonical name

ImaginaryQuadraticField

Date of creation

2013-03-22 17:58:21

Last modified on

2013-03-22 17:58:21

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

8

Author

pahio (2872)

Entry type

Definition

Classification

msc 11R11

Classification

msc 11R04

Synonym

quadratic imaginary field

Synonym

imaginary quadratic number field

Synonym

quadratic imaginary number field

Synonym

imaginary quadratic extension

Related topic

Imaginary

Related topic

UnitsOfQuadraticFields

Related topic

QuadraticImaginaryEuclideanNumberFields

Related topic

ListOfAllImaginaryQuadraticPIDs

Defines

real quadratic field

Generated on Fri Feb 9 15:42:32 2018 by

LaTeXML
lemma for imaginary quadratic fields

lemma for imaginary quadratic fields

For determining the

imaginary quadratic fields

whose

ring of integers

has

unique factorization

, one can use the following

Lemma.

Let

d

be a negative integer with

d

≡

1

(

mod

4

)

,

p

the greatest odd

irreducible

(

http://

planetmath

.org/Irreducible

)

integer

with

p

≦

1

3

⁢

|

d

|

and

q

=

1

4

⁢

(

1

-

d

)

.  In the imaginary quadratic field

ℚ

⁢

(

d

)

, the factorization of integers is unique (

http://planetmath.org/Ufd

) if and only if the integers

t

2

-

t

+

q

(

t

=

1

,

2

,

…

,

p

+

1

2

)

(1)

are irreducible (

http://planetmath.org/Irreducible

) in the field of the

rational numbers

.

The lemma yields the below table:

q

d

=

1

-

4

⁢

q

p

1

2

⁢

(

p

+

1

)

the numbers (1)

1

-

3

1

1

1

2

-

7

1

1

2

3

-

11

1

1

3

5

-

19

1

1

5

11

-

43

3

2

11, 13

17

-

67

3

2

17, 19

41

-

163

7

4

41, 43, 47, 53

References

1

K. Väisälä

:

Lukuteorian ja korkeamman algebran alkeet

.  Tiedekirjasto No. 17. Kustannusosakeyhtiö Otava, Helsinki (1950).

Title

lemma for imaginary quadratic fields

Canonical name

LemmaForImaginaryQuadraticFields

Date of creation

2013-03-22 18:31:23

Last modified on

2013-03-22 18:31:23

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

7

Author

pahio (2872)

Entry type

Theorem

Classification

msc 11R11

Classification

msc 11R04

Related topic

ListOfAllImaginaryQuadraticPIDs

Related topic

ClassNumbersOfImaginaryQuadraticFields

Generated on Fri Feb 9 15:42:35 2018 by

LaTeXML
prime ideal decomposition in quadratic extensions of ℚ

prime ideal decomposition in quadratic extensions of

ℚ

Let

K

be a

quadratic number field

, i.e.

K

=

ℚ

⁢

(

d

)

for
some square-free integer

d

. The

discriminant

of the

extension

is

D

K

/

ℚ

=

{

d

,

if

⁢

d

≡

1

⁢

mod

⁡

4

,

4

⁢

d

,

if

⁢

d

≡

2

,

3

⁢

mod

⁡

4

.

Let

𝒪

K

denote the

ring of integers

of

K

. We have:

𝒪

K

≅

{

ℤ

⊕

1

+

d

2

⁢

ℤ

,

if

⁢

d

≡

1

⁢

mod

⁡

4

,

ℤ

⊕

d

⁢

ℤ

,

if

⁢

d

≡

2

,

3

⁢

mod

⁡

4

.

Prime ideals

of

ℤ

decompose as follows in

𝒪

K

:

Theorem 1

.

Let

p

∈

Z

be a prime.

1.

If

p

∣

d

(divides), then

p

⁢

𝒪

K

=

(

p

,

d

)

2

;

2.

If

d

is odd, then

2

⁢

𝒪

K

=

{

(

2

,

1

+

d

)

2

,

if

⁢

d

≡

3

⁢

mod

⁡

4

,

(

2

,

1

+

d

2

)

⁢

(

2

,

1

-

d

2

)

,

if

⁢

d

≡

1

⁢

mod

⁡

8

,

𝑝𝑟𝑖𝑚𝑒

,

if

⁢

d

≡

5

⁢

mod

⁡

8

.

3.

If

p

≠

2

,

p

does not divide

d

, then

p

⁢

𝒪

K

=

{

(

p

,

n

+

d

)

⁢

(

p

,

n

-

d

)

,

if

⁢

d

≡

n

2

⁢

mod

⁡

p

,

𝑝𝑟𝑖𝑚𝑒

,

if

d

is not a square

⁢

mod

⁡

p

.

References

1

Daniel A.Marcus,

Number Fields

. Springer, New York.

Title

prime ideal decomposition in

quadratic extensions

of

ℚ

Canonical name

PrimeIdealDecompositionInQuadraticExtensionsOfmathbbQ

Date of creation

2013-03-22 13:53:46

Last modified on

2013-03-22 13:53:46

Owner

alozano (2414)

Last modified by

alozano (2414)

Numerical id

7

Author

alozano (2414)

Entry type

Theorem

Classification

msc 11R11

Related topic

CalculatingTheSplittingOfPrimes

Related topic

ExamplesOfPrimeIdealDecompositionInNumberFields

Related topic

PrimeIdealDecompositionInCyclotomicExtensionsOfMathbbQ

Related topic

NumberField

Related topic

SplittingAndRamificationInNumberFieldsAndGaloisExtensions

Generated on Fri Feb 9 15:42:37 2018 by

LaTeXML
quadratic fields that are not isomorphic

quadratic fields that are not isomorphic

Within this entry,

S

denotes the set of all

squarefree

integers

not equal to

1

.

Theorem

.

Let

m

,

n

∈

S

with

m

≠

n

. Then

Q

⁢

(

m

)

and

Q

⁢

(

n

)

are not

isomorphic

(

http://planetmath.org/FieldIsomorphism

).

Proof.

Suppose that

ℚ

⁢

(

m

)

and

ℚ

⁢

(

n

)

are isomorphic. Let

φ

:

ℚ

⁢

(

m

)

→

ℚ

⁢

(

n

)

be a

field isomorphism

. Recall that

field homomorphisms fix prime subfields

. Thus, for every

x

∈

ℚ

,

φ

⁢

(

x

)

=

x

.

Let

a

,

b

∈

ℚ

with

φ

⁢

(

m

)

=

a

+

b

⁢

n

. Since

φ

⁢

(

a

)

=

a

and

φ

is

injective

,

b

≠

0

. Also,

m

=

φ

⁢

(

m

)

=

φ

⁢

(

(

m

)

2

)

=

(

φ

⁢

(

m

)

)

2

=

(

a

+

b

⁢

n

)

2

=

a

2

+

2

⁢

a

⁢

b

⁢

n

+

b

2

⁢

n

. If

a

≠

0

, then

n

=

m

-

a

2

-

b

2

⁢

n

2

⁢

a

⁢

b

∈

ℚ

, a

contradiction

. Thus,

a

=

0

. Therefore,

m

=

b

2

⁢

n

. Since

m

is squarefree,

b

2

=

1

. Hence,

m

=

n

, a contradiction. It follows that

K

and

L

are not isomorphic.
∎

This yields an obvious corollary:

Corollary

.

There are infinitely many distinct

quadratic fields

.

Proof.

Note that there are infinitely many elements of

S

. Moreover, if

m

and

n

are distinct elements of

S

, then

ℚ

⁢

(

m

)

and

ℚ

⁢

(

n

)

are not isomorphic and thus cannot be equal.
∎

Note that the above corollary could have also been obtained by using the result regarding

Galois groups

of finite

abelian extensions

of

ℚ

(

http://planetmath.org/GaloisGroupsOfFiniteAbelianExtensionsOfMathbbQ

). On the other hand, using this result to prove the above corollary can be likened to “using a sledgehammer to kill a housefly”.

Title

quadratic fields that are not isomorphic

Canonical name

QuadraticFieldsThatAreNotIsomorphic

Date of creation

2013-03-22 16:19:44

Last modified on

2013-03-22 16:19:44

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

9

Author

Wkbj79 (1863)

Entry type

Theorem

Classification

msc 11R11

Generated on Fri Feb 9 15:42:46 2018 by

LaTeXML
biquadratic extension

biquadratic extension

A

biquadratic extension

of a field

F

is a

Galois extension

K

of

F

such that

Gal

⁡

(

K

/

F

)

is

isomorphic

to the

Klein 4-group

. It receives its name from the fact that any such

K

is the

compositum

of two distinct

quadratic extensions

of

F

. The name can be somewhat misleading, however, since biquadratic extensions of

F

have exactly three distinct

subfields

that are quadratic extensions of

F

. This is easily seen to be true by the fact that the Klein 4-group has exactly three distinct

subgroups

of order (

http://planetmath.org/OrderGroup

) 2.

Note that, if

α

,

β

∈

F

, then

F

⁢

(

α

,

β

)

is a biquadratic extension of

F

if and only if none of

α

,

β

, and

α

⁢

β

are squares in

F

.

Title

biquadratic extension

Canonical name

BiquadraticExtension

Date of creation

2013-03-22 15:56:21

Last modified on

2013-03-22 15:56:21

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

7

Author

Wkbj79 (1863)

Entry type

Definition

Classification

msc 11R16

Related topic

BiquadraticField

Related topic

BiquadraticEquation2

Generated on Fri Feb 9 15:42:47 2018 by

LaTeXML
Galois group of a biquadratic extension

Galois group of a biquadratic extension

This article proves that

biquadratic extensions

correspond precisely to

Galois extensions

with

Galois group

isomorphic

to the Klein

4

-group

V

4

(at least if the

characteristic

of the

base field

is not

2

). More precisely,

Theorem 1

.

Let

F

be a field of characteristic

≠

2

and

K

a

finite extension

of

F

. Then the following are

equivalent

:

1.

K

=

F

⁢

(

D

1

,

D

2

)

for some

D

1

,

D

2

∈

F

such that none of

D

1

,

D

2

, or

D

1

⁢

D

2

is a square in

F

.

2.

K

is a Galois extension of

F

with

Gal

⁡

(

K

/

F

)

≅

V

4

;

Proof.

Suppose first that condition (1) holds. Then

[

F

(

D

1

)

:

F

]

=

[

F

(

D

2

)

:

F

]

=

2

since neither

D

1

nor

D

2

is a square in

F

. Now obviously

[

K

:

F

]

=

[

F

(

D

1

,

D

2

)

:

F

(

D

1

)

]

[

F

(

D

1

)

:

F

]

≤

4

and so

[

K

:

F

(

D

1

)

]

≤

2

. If

K

=

F

⁢

(

D

1

)

, then

D

2

∈

F

⁢

(

D

1

)

, so

D

2

=

a

+

b

⁢

D

1

and

D

2

=

a

2

+

b

2

⁢

D

1

+

2

⁢

a

⁢

b

⁢

D

1

. Thus

a

=

0

or

b

=

0

. If

b

=

0

, then

D

2

is a square. If

a

=

0

, then

D

1

⁢

D

2

=

b

2

⁢

D

1

2

is a square. In any case, this is a

contradiction

. Thus

K

is a

quadratic extension

of

F

⁢

(

D

1

)

. So

[

K

:

F

]

=

4

. But

K

is the

splitting field

for

(

x

2

-

D

1

)

⁢

(

x

2

-

D

2

)

, since the splitting field must contain both

square roots

, and the

polynomial

obviously splits in

K

, so

G

=

Gal

⁡

(

K

/

F

)

has four

elements

i

⁢

d

σ

:

{

D

1

↦

-

D

1

D

2

↦

D

2

τ

:

{

D

1

↦

D

1

D

2

↦

-

D

2

σ

⁢

τ

:

{

D

1

↦

-

D

1

D

2

↦

-

D

2

and is thus isomorphic to

V

4

.

Now assume that condition (2) holds. Since

Gal

⁡

(

K

/

F

)

≅

V

4

, there must be three intermediate

subfields

E

1

,

E

2

,

E

3

between

F

and

K

of degree

2

over

F

corresponding to the three

subgroups

of

V

4

of order

2

. Thus each of these is a quadratic extension. Suppose

E

1

=

F

⁢

(

D

1

)

,

E

2

=

F

⁢

(

D

2

)

where neither

D

1

nor

D

2

is a square in

F

. The fact that

E

1

≠

E

2

implies as above that

D

1

⁢

D

2

is also not a square in

F

(in fact

E

3

=

F

⁢

(

D

1

⁢

D

2

)

. Thus

E

1

⁢

E

2

⊋

E

1

,

E

2

, and is of degree

4

over

F

, so

K

=

E

1

⁢

E

2

=

F

⁢

(

D

1

,

D

2

)

.

Title

Galois group of a biquadratic extension

Canonical name

GaloisGroupOfABiquadraticExtension

Date of creation

2013-03-22 17:44:06

Last modified on

2013-03-22 17:44:06

Owner

rm50 (10146)

Last modified by

rm50 (10146)

Numerical id

5

Author

rm50 (10146)

Entry type

Theorem

Classification

msc 11R16

Generated on Fri Feb 9 15:42:53 2018 by

LaTeXML
pure cubic field

pure cubic field

A

pure cubic field

is an

extension

of

ℚ

of the form

ℚ

⁢

(

n

3

)

for some

n

∈

ℤ

such that

n

3

∉

ℚ

. If

n

<

0

, then

n

3

=

-

|

n

|

3

=

-

|

n

|

3

, causing

ℚ

⁢

(

n

3

)

=

ℚ

⁢

(

|

n

|

3

)

. Thus, without loss of generality, it may be assumed that

n

>

1

.

Note that no pure cubic field is Galois (

http://planetmath.org/GaloisExtension

) over

ℚ

. For if

n

∈

ℤ

is

cubefree

with

|

n

|

≠

1

, then

x

3

-

n

is its

minimal polynomial

over

ℚ

. This

polynomial

factors as

(

x

-

n

3

)

⁢

(

x

2

+

x

⁢

n

3

+

n

2

3

)

over

K

=

ℚ

⁢

(

|

n

|

3

)

. The

discriminant

(

http://planetmath.org/PolynomialDiscriminant

) of

x

2

+

x

⁢

n

3

+

n

2

3

is

(

n

3

)

2

-

4

⁢

(

1

)

⁢

(

n

2

3

)

=

n

2

3

-

4

⁢

n

2

3

=

-

3

⁢

n

2

3

. Since the of

x

2

+

x

⁢

n

3

+

n

2

3

is

negative

, it does not factor in

ℝ

. Note that

K

⊆

ℝ

. Thus,

x

3

-

n

has a root (

http://planetmath.org/Root

) in

K

but does not split completely in

K

.

Note also that pure cubic fields are real cubic fields with exactly one

real embedding

. Thus, a possible method of determining all of the units of pure cubic fields is outlined in the entry regarding

units of real cubic fields with exactly one real embedding

.

Title

pure cubic field

Canonical name

PureCubicField

Date of creation

2013-03-22 16:02:19

Last modified on

2013-03-22 16:02:19

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

14

Author

Wkbj79 (1863)

Entry type

Definition

Classification

msc 11R16

Generated on Fri Feb 9 15:43:02 2018 by

LaTeXML
using the primitive element of biquadratic field

using the primitive element of biquadratic field

Let

m

and

n

be two distinct

squarefree

integers

≠

1

.  We want to express their

square roots

as

polynomials

of

α

:=

m

+

n

(1)

with

rational

coefficients.

If

α

is cubed (

http://planetmath.org/CubeOfANumber

), the result no terms with

m

⁢

n

:

α

3

=

(

m

)

3

+

3

⁢

(

m

)

2

⁢

n

+

3

⁢

m

⁢

(

n

)

2

+

(

n

)

3

=

m

⁢

m

+

3

⁢

m

⁢

n

+

3

⁢

n

⁢

m

+

n

⁢

n

=

(

m

+

3

⁢

n

)

⁢

m

+

(

3

⁢

m

+

n

)

⁢

n

Thus, if we subtract from this the product

(

3

⁢

m

+

n

)

⁢

α

, the

n

term vanishes:

α

3

-

(

3

⁢

m

+

n

)

⁢

α

=

(

-

2

⁢

m

+

2

⁢

n

)

⁢

m

Dividing this equation by

-

2

⁢

m

+

2

⁢

n

(

≠

0

) yields

m

=

α

3

-

(

3

⁢

m

+

n

)

⁢

α

2

⁢

(

-

m

+

n

)

.

(2)

Similarly, we have

n

=

α

3

-

(

m

+

3

⁢

n

)

⁢

α

2

⁢

(

m

-

n

)

.

(3)

The (2) and (3) may be interpreted as such polynomials as intended.

Multiplying the equations (2) and (3) we obtain a corresponding for the square root of

m

⁢

n

which also lies in the quartic field

ℚ

⁢

(

m

,

n

)

=

ℚ

⁢

(

m

+

n

)

:

m

⁢

n

=

α

6

-

4

⁢

(

m

+

n

)

⁢

α

4

+

(

3

⁢

m

2

+

10

⁢

m

⁢

n

+

3

⁢

n

2

)

⁢

α

2

4

⁢

(

-

m

2

+

2

⁢

m

⁢

n

-

n

2

)

For example, in the special case

m

:=

2

,

n

:=

3

we have

2

=

α

3

-

9

⁢

α

2

,

3

=

-

α

3

-

11

⁢

α

2

,

6

=

-

α

6

+

20

⁢

α

4

-

99

⁢

α

2

4

.

Remark.

The sum (1) of two square roots of

positive

squarefree integers is always

irrational

, since in the contrary case, the equation (3) would say that

n

would be rational; this has been proven impossible here (

http://planetmath.org/SquareRootOf2IsIrrationalProof

).

Title

using the

primitive element of biquadratic field

Canonical name

UsingThePrimitiveElementOfBiquadraticField

Date of creation

2013-03-22 17:54:25

Last modified on

2013-03-22 17:54:25

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

12

Author

pahio (2872)

Entry type

Application

Classification

msc 11R16

Synonym

expressing two square roots with their sum

Synonym

irrational sum of square roots

Related topic

BinomialTheorem

Generated on Fri Feb 9 15:43:05 2018 by

LaTeXML