quadratic imaginary norm-Euclidean number fields

quadratic imaginary norm-Euclidean number fields

Theorem 1.

The

imaginary quadratic fields

ℚ

⁢

(

d

)

with

d

=

-

1

,

-

2

,

-

3

,

-

7

,

-

11

are

norm-Euclidean number fields

.

Proof.

1

∘

.

d

≢

1

⁢

(

mod

⁢

4

)

,  i.e.

d

=

-

1

or

d

=

-

2

.  Any element

γ

of the field

ℚ

⁢

(

d

)

has the

canonical form

γ

=

c

0

+

c

1

⁢

d

where

c

0

,

c

1

∈

ℚ

.  We may write

γ

=

(

p

+

r

)

+

(

q

+

s

)

⁢

d

, where

p

is the

rational integer

nearest to

c

0

and

q

the one nearest to

c

1

.  So

|

r

|

≤

1

2

,

|

s

|

≤

1

2

.  Thus we may write

γ

=

(

p

+

q

⁢

d

)

⏟

ϰ

+

(

r

+

s

⁢

d

)

⏟

δ

,

where

ϰ

is an

integer

of the field.  We then can

0

≤

N

⁢

(

δ

)

=

(

r

+

s

⁢

d

)

⁢

(

r

-

s

⁢

d

)

=

r

2

-

s

2

⁢

d

=

r

2

+

s

2

⁢

|

d

|

≤

(

1

2

)

2

+

2

⁢

(

1

2

)

2

=

3

4

<

1

,

and therefore

|

N

⁢

(

δ

)

|

<

1

.  According to the theorem 1 in the parent entry (

http://planetmath.org/EuclideanNumberField

),

ℚ

⁢

(

-

1

)

and

ℚ

⁢

(

-

2

)

are norm-Euclidean number fields.

2

∘

.

d

≡

1

⁢

(

mod

⁢

4

)

,  i.e.

d

∈

{

-

3

,

-

7

,

-

11

}

.  The

algebraic integers

of

ℚ

⁢

(

d

)

have now the canonical form

a

+

b

⁢

d

2

with

2

|

a

-

b

.  Let

γ

=

c

0

+

c

1

⁢

d

where

c

0

,

c

1

∈

ℚ

be an arbitrary element of the field.  Choose the rational integer

q

such that

q

2

is as close to

c

1

as possible, i.e.

c

1

=

q

2

+

s

with

|

s

|

≤

1

4

,  and the rational integer

t

such that

q

2

+

t

is as close to

c

0

as possible; then

c

0

=

q

+

2

⁢

t

2

+

r

=

p

2

+

r

with

|

r

|

≤

1

2

.  Then we can write

γ

=

p

2

+

r

+

(

q

2

+

s

)

⁢

d

=

p

+

q

⁢

d

2

⏟

ϰ

+

(

r

+

s

⁢

d

)

⏟

δ

.

The number

ϰ

is an integer of the field, since

p

-

q

=

2

⁢

t

≡

0

⁢

(

mod

⁢

2

)

.  We get the estimation

0

≤

N

⁢

(

δ

)

=

r

2

+

s

2

⁢

|

d

|

≤

(

1

2

)

2

+

11

⁢

(

1

4

)

2

=

15

16

<

1

,

so

|

N

⁢

(

δ

)

|

<

1

.  Thus the fields in question are norm-Euclidean number fields.

Theorem 2.

The only

quadratic imaginary norm-Euclidean number fields

ℚ

⁢

(

d

)

are the ones in which

d

=

-

1

,

-

2

,

-

3

,

-

7

,

-

11

.

Proof.

Let

d

be any other

negative

(

square-free

) rational integer than the above mentioned ones.

1

∘

.

d

≢

1

⁢

(

mod

⁢

4

)

.  The integers of

ℚ

⁢

(

d

)

are

a

+

b

⁢

d

where

a

,

b

∈

ℤ

.  We show that there is a number

γ

that can not be expressed in the form

γ

=

ϰ

+

δ

with

ϰ

an integer of the field and

|

N

⁢

(

δ

)

|

<

1

.  Assume that

γ

:=

1

2

⁢

d

=

ϰ

+

δ

where

ϰ

=

a

+

b

⁢

d

is an integer of the field (

a

,

b

∈

ℤ

).  Then

δ

=

γ

-

ϰ

=

-

a

+

(

1

2

-

b

)

⁢

d

and

N

⁢

(

δ

)

=

|

a

|

2

+

|

d

|

⋅

|

1

2

-

b

|

2

.  Because

b

cannot be 0, we have

|

1

2

-

b

|

≥

1

2

and thus

|

N

⁢

(

δ

)

|

≥

0

+

|

d

|

⁢

(

1

2

)

2

=

|

d

|

4

≥

5

4

>

1

.

Therefore

ℚ

⁢

(

d

)

can not be a norm-Euclidean number field
(

d

=

-

5

,

-

6

,

-

10

and so on).

2

∘

.

d

≡

1

⁢

(

mod

⁢

4

)

.  Now

|

d

|

≥

15

.  The integers of

ℚ

⁢

(

d

)

have the form

ϰ

=

a

+

b

⁢

d

2

with

2

|

a

-

b

.  Suppose that

γ

=

1

4

+

1

4

⁢

d

=

ϰ

+

δ

.  Then

δ

=

γ

-

ϰ

=

(

1

4

-

a

2

)

+

(

1

4

-

b

2

)

⁢

d

and

|

N

⁢

(

δ

)

|

≥

|

1

4

-

a

2

|

2

+

|

d

|

⋅

|

1

4

-

b

2

|

2

≥

(

1

4

)

2

+

15

⁢

(

1

4

)

2

=

1

.

So also these fields

ℚ

⁢

(

d

)

are not norm-Euclidean number fields.

Remark.

The

rings of integers

of the imaginary quadratic fields of the above theorems are thus PID’s.  There are, in

addition

, four other imaginary quadratic fields which are not norm-Euclidean but anyway their rings of integers are PID’s (see

lemma for imaginary quadratic fields

,

class numbers of imaginary quadratic fields

,

unique factorization and ideals in ring of integers

,

divisor theory

).

Title

quadratic imaginary norm-Euclidean number fields

Canonical name

QuadraticImaginaryNormEuclideanNumberFields

Date of creation

2013-03-22 16:52:32

Last modified on

2013-03-22 16:52:32

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

14

Author

pahio (2872)

Entry type

Theorem

Classification

msc 13F07

Classification

msc 11R21

Classification

msc 11R04

Synonym

imaginary quadratic Euclidean number fields

Synonym

imaginary Euclidean quadratic fields

Related topic

EuclideanNumberField

Related topic

ImaginaryQuadraticField

Related topic

ClassNumbersOfImaginaryQuadraticFields

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