basis of ideal in algebraic number field

basis of ideal in algebraic number field

Theorem

.

Let

𝒪

K

be the

maximal order

of the

algebraic number field

K

of degree

n

.  Every ideal

𝔞

of

𝒪

K

has a

basis

, i.e. there are in

𝔞

the

linearly independent

numbers

α

1

,

α

2

,

…

,

α

n

such that the numbers

m

1

⁢

α

1

+

m

2

⁢

α

2

+

…

+

m

n

⁢

α

n

,

where the

m

i

’s run all

rational integers

, form precisely all numbers of

𝔞

.  One has also

𝔞

=

(

α

1

,

α

2

,

…

,

α

n

)

,

i.e. the basis of the ideal can be taken for the system of

generators

of the ideal.

Since

{

α

1

,

α

2

,

…

,

α

n

}

is a basis of the

field extension

K

/

ℚ

, any element of

𝔞

is uniquely

expressible

in the form

m

1

⁢

α

1

+

m

2

⁢

α

2

+

…

+

m

n

⁢

α

n

.

It may be proven that all bases of an ideal

𝔞

have the same

discriminant

Δ

⁢

(

α

1

,

α

2

,

…

,

α

n

)

, which is an

integer

; it is called the

discriminant of the ideal

.  The discriminant of the ideal has the minimality property, that if

β

1

,

β

2

,

…

,

β

n

are some elements of

𝔞

, then

Δ

⁢

(

β

1

,

β

2

,

…

,

β

n

)

≧

Δ

⁢

(

α

1

,

α

2

,

…

,

α

n

)

or

Δ

⁢

(

β

1

,

β

2

,

…

,

β

n

)

=

0

But if

Δ

⁢

(

β

1

,

β

2

,

…

,

β

n

)

=

Δ

⁢

(

α

1

,

α

2

,

…

,

α

n

)

,  then also the

β

i

’s form a basis of the ideal

𝔞

.

Example.

The integers of the

quadratic field

ℚ

⁢

(

2

)

are

l

+

m

⁢

2

with

l

,

m

∈

ℤ

.  Determine a basis

{

α

1

,

α

2

}

and the discriminant of the ideal
a)

(

6

-

6

⁢

2

,

9

+

6

⁢

2

)

,  b)

(

1

-

2

⁢

2

)

.

a) The ideal may be seen to be the

principal ideal

(

3

)

, since the both generators are of the form

(

l

+

m

⁢

2

)

⋅

3

and on the other side,

3

=

0

⋅

(

6

-

6

⁢

2

)

+

(

3

-

2

⁢

2

)

⁢

(

9

+

6

⁢

2

)

.  Accordingly, any element of the ideal are of the form

(

m

1

+

m

2

⁢

2

)

⋅

3

=

m

1

⋅

3

+

m

2

⋅

3

⁢

2

where

m

1

and

m

2

are rational integers.  Thus we can infer that

α

1

=

3

,

α

2

=

3

⁢

2

is a basis of the ideal concerned.  So its discriminant is

Δ

⁢

(

α

1

,

α

2

)

=

|

3

3

⁢

2

3

-

3

⁢

2

|

2

=

648

.

b) All elements of the ideal

(

1

-

2

⁢

2

)

have the form

α

=

(

a

+

b

⁢

2

)

⁢

(

1

-

2

⁢

2

)

=

(

a

-

4

⁢

b

)

+

(

b

-

2

⁢

a

)

⁢

2

with

⁢

a

,

b

∈

ℤ

.

(1)

Especially the rational integers of the ideal

satisfy

b

-

2

⁢

a

=

0

,  when

b

=

2

⁢

a

and thus

α

=

a

-

4

⋅

2

⁢

a

=

-

7

⁢

a

.  This means that in the

presentation

α

=

m

1

⁢

α

1

+

m

2

⁢

α

2

we can assume

α

1

to be

7

.  Now the

rational

portion

a

-

4

⁢

b

in the form (1) of

α

should be splitted into two parts so that the first would be always

divisible

by 7 and the second by

b

-

2

⁢

a

, i.e.

a

-

4

⁢

b

=

7

⁢

m

1

+

(

b

-

2

⁢

a

)

⁢

x

; this

equation

may be written also as

(

2

⁢

x

+

1

)

⁢

a

-

(

x

+

4

)

⁢

b

=

7

⁢

m

1

.

By experimenting, one finds the simplest value

x

=

3

,  another would be

x

=

10

.  The first of these yields

α

=

7

⁢

(

a

-

b

)

+

(

b

-

2

⁢

a

)

⁢

(

3

+

2

)

=

m

1

⋅

7

+

m

2

⁢

(

3

+

2

)

,

i.e. we have the basis

α

1

=

7

,

α

2

=

3

+

2

.

The second alternative

x

=

10

similarly would give

α

1

=

7

,

α

2

=

10

+

2

.

For both alternatives,

Δ

⁢

(

α

1

,

α

2

)

=

392

.

Title

basis of ideal in algebraic number field

Canonical name

BasisOfIdealInAlgebraicNumberField

Date of creation

2013-03-22 17:51:15

Last modified on

2013-03-22 17:51:15

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

9

Author

pahio (2872)

Entry type

Theorem

Classification

msc 12F05

Classification

msc 11R04

Classification

msc 06B10

Synonym

basis of ideal in number field

Related topic

IntegralBasis

Related topic

IdealNorm

Related topic

AlgebraicNumberTheory

Defines

basis of ideal

Defines

ideal basis

Defines

discriminant of the ideal

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