integral basis of quadratic field

integral basis of quadratic field

Let

m

be a

squarefree

integer

≠

1

. All numbers of the

quadratic field

ℚ

⁢

(

m

)

may be written in the
form

α

=

j

+

k

⁢

m

l

,

(1)

where

j

,

k

,

l

are integers with

gcd

⁡

(

j

,

k

,

l

)

=

1

and

l

>

0

.  Then

α

(and its

algebraic conjugate

α

′

=

j

-

k

⁢

m

l

)

satisfies

the equation

x

2

+

p

⁢

x

+

q

=

0

,

(2)

where

p

=

-

2

⁢

j

l

,

q

=

j

2

-

k

2

⁢

m

l

2

.

(3)

We will find out when the number (1) is an

algebraic integer

, i.e. when the coefficients

p

and

q

are

rational integers

.

Naturally,

p

and

q

are integers always when

l

=

1

.  We
suppose now that

l

>

1

.  The latter of the equations (3)
says that

q

can be integer only when

(

gcd

(

j

,

l

)

)

2

=

gcd

(

j

2

,

l

2

)

∣

k

2

m

(see divisibility in rings).  Because

gcd

⁡

(

j

,

k

,

l

)

=

1

,
we have by Euclid’s lemma that

gcd

(

j

,

l

)

∣

m

.  Since

m

is squarefree, we infer that

gcd

⁡

(

j

,

l

)

=

1

.

(4)

In order that also

p

were an integer, the former of the
equations (3) implies that

l

=

2

.

So, by the latter of the equations (3),

4

∣

j

2

-

k

2

m

,
i.e.

k

2

⁢

m

≡

j

2

(

mod

4

)

.

(5)

Since by (4),

gcd

⁡

(

j

,

2

)

=

1

,  the integer

j

has to be odd.  In order that (5) would be valid, also

k

must be odd.  Therefore,

j

2

≡

1

(

mod

4

)

and

k

2

≡

1

(

mod

4

)

,  and thus (5) changes to

m

≡

1

(

mod

4

)

.

(6)

If we

conversely

assume (6) and that

j

,

k

are odd and

l

=

2

, then (5) is true,

p

,

q

are integers and accordingly (1) is an algebraic integer.

We have now obtained the following result:

•

When

m

≢

1

(

mod

4

)

,  the integers of the field

ℚ

⁢

(

m

)

are

a

+

b

⁢

m

where

a

,

b

are arbitrary rational integers;

•

when

m

≡

1

(

mod

4

)

,  in to the numbers

a

+

b

⁢

m

, also the numbers

j

+

k

⁢

m

2

,

with

j

,

k

arbitrary

odd integers

, are integers of the field.

Then, it may be easily inferred the

Theorem

.

If we denote

ω

:=

{

1

+

m

2

when

⁢

m

≡

1

(

mod

4

)

,

m

when

⁢

m

≢

1

(

mod

4

)

,

then any integer of the quadratic field

ℚ

⁢

(

m

)

may be expressed in the form

a

+

b

⁢

ω

,

where

a

and

b

are uniquely determined rational integers.
Conversely, every number of this form is an integer of the field.  One says that 1 and

ω

form an

integral basis

of the field.

References

1

K. Väisälä

:

Lukuteorian ja korkeamman algebran alkeet

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

Title

integral basis of quadratic field

Canonical name

IntegralBasisOfQuadraticField

Date of creation

2014-02-27 10:24:31

Last modified on

2014-02-27 10:24:31

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

11

Author

pahio (2872)

Entry type

Derivation

Classification

msc 11R04

Synonym

canonical basis of quadratic field

Synonym

quadratic integers

Related topic

PropertiesOfQuadraticEquation

Related topic

Gcd

Related topic

ExamplesOfRingOfIntegersOfANumberField

Related topic

SomethingRelatedToFundamentalUnits

Related topic

CanonicalBasis

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