norm-Euclidean number field

norm-Euclidean number field

Definition.

An

algebraic number field

K

is a

norm-Euclidean number field

, if for every pair

(

α

,

β

)

of the

integers

(

http://planetmath.org/AlgebraicInteger

) of

K

, where

β

≠

0

,  there exist

ϰ

and

ϱ

of the field such that

α

=

ϰ

⁢

β

+

ϱ

,

|

N

⁢

(

ϱ

)

|

<

|

N

⁢

(

β

)

|

.

Here

N

means the norm

function

in

K

.

Theorem

1.

A field

K

is norm-Euclidean if and only if each number

γ

of

K

is in the form

γ

=

ϰ

+

δ

(1)

where

ϰ

is an of the field and

|

N

⁢

(

δ

)

|

<

1

.

Proof.

First assume the condition (1).  Let

α

and

β

be integers of

K

,

β

≠

0

.  Then there are the numbers

ϰ

,

δ

∈

K

such that

ϰ

is integer and

α

β

=

ϰ

+

δ

,

|

N

⁢

(

δ

)

|

<

1

.

Thus we have

α

=

ϰ

⁢

β

+

β

⁢

δ

=

ϰ

⁢

β

+

ϱ

.

Here

ϱ

=

β

⁢

δ

is integer, since

α

and

ϰ

⁢

β

are integers.  We also have

|

N

⁢

(

ϱ

)

|

=

|

N

⁢

(

β

)

|

⋅

|

N

⁢

(

δ

)

|

<

|

N

⁢

(

β

)

|

⋅

1

=

|

N

⁢

(

β

)

|

.

Accordingly,

K

is a norm-Euclidean number field.
Secondly assume that

K

is norm-Euclidean.  Let

γ

be an arbitrary element of the field.  We can determine (

http://planetmath.org/MultiplesOfAnAlgebraicNumber

) a

rational integer

m

(

≠

0

)

such that

m

⁢

γ

is an

algebraic integer

of

K

.  The

assumption

guarantees the integers

ϰ

,

ϱ

of

K

such that

m

⁢

γ

=

ϰ

⁢

m

+

ϱ

,

N

⁢

(

ϱ

)

<

N

⁢

(

m

)

.

Thus

γ

=

m

⁢

γ

m

=

ϰ

+

ϱ

m

,

|

N

⁢

(

ϱ

m

)

|

=

|

N

⁢

(

ϱ

)

|

|

N

⁢

(

m

)

|

<

1

,

Q.E.D.

Theorem 2.

In a norm-Euclidean number field, any two non-zero have a

greatest common divisor

.

Proof.

We recall that the

greatest common divisor

of two elements of a

commutative ring

means such a common

divisor

of the

elements

that it is

divisible

by each common divisor of the elements.  Let now

ϱ

0

and

ϱ

1

be two algebraic integers of a norm-Euclidean number field

K

.  According the definition there are the integers

ϰ

i

and

ϱ

i

of

K

such that

{

ϱ

0

=

ϰ

2

⁢

ϱ

1

+

ϱ

2

,

|

N

⁢

(

ϱ

2

)

|

<

|

N

⁢

(

ϱ

1

)

|

ϱ

1

=

ϰ

3

⁢

ϱ

2

+

ϱ

3

,

|

N

⁢

(

ϱ

3

)

|

<

|

N

⁢

(

ϱ

2

)

|

ϱ

2

=

ϰ

4

⁢

ϱ

3

+

ϱ

4

,

|

N

⁢

(

ϱ

4

)

|

<

|

N

⁢

(

ϱ

3

)

|

⋯

⁢

⋯

ϱ

n

-

2

=

ϰ

n

⁢

ϱ

n

-

1

+

ϱ

n

,

|

N

⁢

(

ϱ

n

)

|

<

|

N

⁢

(

ϱ

n

-

1

)

|

ϱ

n

-

1

=

ϰ

n

+

1

⁢

ϱ

n

+

0

,

The ends to the

remainder

0, because the numbers

|

N

⁢

(

ϱ

i

)

|

form a descending

sequence

of non-negative rational integers — see the entry

norm and trace of algebraic number

.  As in the Euclid’s algorithm in

ℤ

, one sees that the last divisor

ϱ

n

is one greatest common divisor of

ϱ

0

and

ϱ

1

.  N.B. that

ϱ

0

and

ϱ

1

may have an

infinite

amount of their greatest common divisors, depending the amount of the units in

K

.

Remark.

The

ring of integers

of any norm-Euclidean number field is a

unique factorization domain

and thus all ideals of the ring are

principal ideals

.  But not all algebraic number fields with ring of integers a UFD (

http://planetmath.org/UFD

) are norm-Euclidean, e.g.

ℚ

⁢

(

14

)

.

Theorem 3.

The only norm-Euclidean

quadratic fields

ℚ

⁢

(

d

)

are those with

d

∈

{

-

11

,

-

7

,

-

3

,

-

2

,

-

1

,

2

,

3

,

5

,

6

,

7

,

11

,

13

,

17

,

19

,

21

,

29

,

33

,

37

,

41

,

57

,

73

}

.

Title

norm-Euclidean number field

Canonical name

NormEuclideanNumberField

Date of creation

2013-03-22 16:52:26

Last modified on

2013-03-22 16:52:26

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

17

Author

pahio (2872)

Entry type

Topic

Classification

msc 13F07

Classification

msc 11R21

Classification

msc 11R04

Related topic

EuclideanValuation

Related topic

QuadraticImaginaryEuclideanNumberFields

Related topic

ListOfAllImaginaryQuadraticPIDs

Related topic

EuclideanField

Related topic

AlgebraicNumberTheory

Related topic

MixedFraction

Defines

norm-Euclidean

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