canonical form of element of number field

canonical form of element of number field

Theorem

.

Let

ϑ

be an

algebraic number

of degree (

http://planetmath.org/DegreeOfAnAlgebraicNumber

)

n

.  Any

element

α

of the

algebraic number field

ℚ

⁢

(

ϑ

)

may be uniquely expressed in the

canonical form

α

=

c

0

+

c

1

⁢

ϑ

+

c

2

⁢

ϑ

2

+

…

+

c

n

-

1

⁢

ϑ

n

-

1

(1)

where the numbers

c

i

are

rational

.

Proof.

We start from the fact that

ℚ

⁢

(

ϑ

)

consists of all

expressions

formed of

ϑ

and rational numbers using

arithmetic operations

(no

divisor

(

http://planetmath.org/

Division

) must vanish); such expressions lead always to the form

α

=

a

⁢

(

ϑ

)

b

⁢

(

ϑ

)

(2)

where the

numerator

and the denominator are

polynomials

in

ϑ

with rational

coefficients

(which can, in fact, be chosen

integers

).

So, let

α

in (2) an arbitrary element of the field

ℚ

⁢

(

ϑ

)

.  Denote by

f

⁢

(

x

)

the

minimal polynomial

of

ϑ

over

ℚ

.  Since

b

⁢

(

ϑ

)

≠

0

,  the polynomial

f

⁢

(

x

)

does not divide (

http://planetmath.org/DivisibilityInRings

)

b

⁢

(

x

)

, and since

f

⁢

(

x

)

is

irreducible

(

http://planetmath.org/IrreduciblePolynomial2

), the

greatest common divisor

(

http://planetmath.org/PolynomialRingOverFieldIsEuclideanDomain

) of

f

⁢

(

x

)

and

b

⁢

(

x

)

is a

constant polynomial

, which can be normed to 1.  Thus there exist the polynomials

φ

⁢

(

x

)

and

ψ

⁢

(

x

)

of the ring

ℚ

⁢

[

x

]

such that

φ

⁢

(

x

)

⁢

f

⁢

(

x

)

+

ψ

⁢

(

x

)

⁢

b

⁢

(

x

)

≡

1

.

Especially

φ

⁢

(

ϑ

)

⁢

f

⁢

(

ϑ

)

⏟

=

0

+

ψ

⁢

(

ϑ

)

⁢

b

⁢

(

ϑ

)

=

1

,

whence

1

b

⁢

(

ϑ

)

=

ψ

⁢

(

ϑ

)

and consequently

α

=

a

⁢

(

ϑ

)

b

⁢

(

ϑ

)

=

a

⁢

(

ϑ

)

⁢

ψ

⁢

(

ϑ

)

:=

ψ

1

⁢

(

ϑ

)

.

Hence,

α

is a polynomial in

ϑ

with rational coefficients.

Let now

ψ

1

⁢

(

x

)

=

q

⁢

(

x

)

⁢

f

⁢

(

x

)

+

r

⁢

(

x

)

with deg

⁢

(

r

)

<

deg

⁢

(

f

)

=

n

.

Denote

r

⁢

(

x

)

:=

c

0

+

c

1

⁢

x

+

…

+

c

n

-

1

⁢

x

n

-

1

∈

ℚ

⁢

[

x

]

.

It follows that

α

=

r

⁢

(

ϑ

)

=

c

0

+

c

1

⁢

ϑ

+

…

+

c

n

-

1

⁢

ϑ

n

-

1

,

whence (1) is true.

Suppose that we had also

α

=

s

⁢

(

ϑ

)

=

d

0

+

d

1

⁢

ϑ

+

…

+

d

n

-

1

⁢

ϑ

n

-

1

with every

d

i

rational.  This implies that

(

c

n

-

1

-

d

n

-

1

)

⁢

ϑ

n

-

1

+

…

+

(

c

1

-

d

1

)

⁢

ϑ

+

(

c

0

-

d

0

)

=

0

,

i.e. that

ϑ

satisfies

the

equation

(

c

n

-

1

-

d

n

-

1

)

⁢

x

n

-

1

+

…

+

(

c

1

-

d

1

)

⁢

x

+

(

c

0

-

d

0

)

=

0

with rational coefficients and degree less than

n

.  Because the degree of

ϑ

is

n

, it is possible only if all

differences

c

i

-

d

i

vanish.  Thus

d

0

=

c

0

,

d

1

=

c

1

,

…

,

d

n

-

1

=

c

n

-

1

,

i.e. the (1) is unique.

Note 1.

The polynomial

c

0

+

c

1

⁢

x

+

…

+

c

n

-

1

⁢

x

n

-

1

is called the

canonical polynomial

of the algebraic number

α

with respect to the

primitive element

(

http://planetmath.org/SimpleFieldExtension

)

ϑ

.

Note 2.

The theorem allows to denote the field

ℚ

⁢

(

ϑ

)

similarly as

polynomial rings

:

ℚ

⁢

[

ϑ

]

.

Note 3.

When allowed, unlike in (1), higher powers of the primitive element

ϑ

(whose minimal polynomial is

x

n

+

a

1

⁢

x

n

-

1

+

…

+

a

n

), one may unlimitedly write different sum of

α

, e.g.

α

=

(

c

0

+

c

1

⁢

ϑ

+

…

+

c

n

-

1

⁢

ϑ

n

-

1

)

+

(

ϑ

n

+

a

1

⁢

ϑ

n

-

1

+

…

+

a

n

)

=

(

c

0

+

a

n

)

+

…

+

(

c

n

-

1

+

a

1

)

⁢

ϑ

n

-

1

+

ϑ

n

.

Title

canonical form of element of number field

Canonical

name

CanonicalFormOfElementOfNumberField

Date of creation

2013-03-22 19:08:00

Last modified on

2013-03-22 19:08:00

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

17

Author

pahio (2872)

Entry type

Theorem

Classification

msc 11R04

Related topic

Canonical

Related topic

SimpleFieldExtension

Related topic

CanonicalBasis

Related topic

IntegralBasis

Defines

canonical form

Defines

canonical form in number field

Defines

canonical polynomial

Generated on Fri Feb 9 15:40:56 2018 by

LaTeXML