polynomial ring over a field

polynomial ring over a field

Theorem

.

The

polynomial ring

over a field is a

Euclidean domain

.

Proof.

Let

K

⁢

[

X

]

be the polynomial ring over a field

K

in the

indeterminate

X

.  Since

K

is an

integral domain

and any

polynomial ring over integral domain

is an integral domain, the ring

K

⁢

[

X

]

is an integral domain.

The degree

ν

⁢

(

f

)

, defined for every

f

in

K

⁢

[

X

]

except the

zero polynomial

,

satisfies

the requirements of a

Euclidean valuation

in

K

⁢

[

X

]

.  In fact, the degrees of polynomials are non-negative

integers

.  If

f

and

g

belong to

K

⁢

[

X

]

and the latter of them is not the zero polynomial, then, as is well known, the

long division

f

/

g

gives two unique

polynomials

q

and

r

in

K

⁢

[

X

]

such that

f

=

q

⁢

g

+

r

,

where

ν

⁢

(

r

)

<

ν

⁢

(

g

)

or

r

is the zero polynomial.  The second property usually required for the Euclidean valuation, is justified by

ν

⁢

(

f

⁢

g

)

=

ν

⁢

(

f

)

+

ν

⁢

(

g

)

≧

ν

⁢

(

f

)

.

The theorem implies, similarly as in the ring

ℤ

of the integers, that one can perform in

K

⁢

[

X

]

a Euclid’s algorithm which yields a

greatest common divisor

of two polynomials.  Performing several Euclid’s algorithms one obtains a gcd of many polynomials; such a gcd is always in the same polynomial ring

K

⁢

[

X

]

.

Let

d

be a greatest common divisor of certain polynomials.  Then apparently also

k

⁢

d

, where

k

is any non-zero element of

K

, is a gcd of the same polynomials.  They do not have other gcd’s than

k

⁢

d

, for if

d

′

is an arbitrary gcd of them, then

d

′

∣

d

and

d

∣

d

′

,

i.e.

d

and

d

′

are

associates

in the ring

K

⁢

[

X

]

and thus

d

′

is gotten from

d

by

multiplication

by an element of the field

K

.  So we can write the

Corollary 1.

The greatest common divisor of polynomials in the ring

K

⁢

[

X

]

is unique up to multiplication by a non-zero element of the field

K

. The monic (

http://planetmath.org/Monic2

) gcd of polynomials is unique.

If the monic gcd of two polynomials is 1, they may be called

coprime

.

Using the Euclid’s algorithm as in

ℤ

, one can prove the

Corollary 2.

If

f

and

g

are two non-zero polynomials in

K

⁢

[

X

]

, this ring contains such polynomials

u

and

v

that

gcd

⁡

(

f

,

g

)

=

u

⁢

f

+

v

⁢

g

and especially, if

f

and

g

are coprime, then

u

and

v

may be chosen such that

u

⁢

f

+

v

⁢

g

=

1

.

Corollary 3.

If a

product

of polynomials in

K

⁢

[

X

]

is

divisible

by an

irreducible polynomial

of

K

⁢

[

X

]

, then at least one factor (

http://planetmath.org/Product

) of the product is divisible by the irreducible polynomial.

Corollary 4.

A polynomial ring over a field is always a

principal ideal domain

.

Corollary 5.

The factorisation of a non-zero polynomial, i.e. the of the polynomial as product of irreducible polynomials, is unique up to

constant

factors in each polynomial ring

K

⁢

[

X

]

over a field

K

containing the polynomial.  Especially,

K

⁢

[

X

]

is a UFD.

Example.

The factorisations of the

trinomial

X

4

-

X

2

-

2

into monic

irreducible

prime factors

are

(

X

2

-

2

)

⁢

(

X

2

+

1

)

in

ℚ

⁢

[

X

]

,

(

X

2

-

2

)

⁢

(

X

+

i

)

⁢

(

X

-

i

)

in

ℚ

⁢

(

i

)

⁢

[

X

]

,

(

X

+

2

)

⁢

(

X

-

2

)

⁢

(

X

2

+

1

)

in

ℚ

⁢

(

2

)

⁢

[

X

]

,

(

X

+

2

)

⁢

(

X

-

2

)

⁢

(

X

+

i

)

⁢

(

X

-

i

)

in

ℚ

⁢

(

2

,

i

)

⁢

[

X

]

.

Title

polynomial ring over a field

Canonical name

PolynomialRingOverAField

Date of creation

2013-03-22 17:42:55

Last modified on

2013-03-22 17:42:55

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

14

Author

pahio (2872)

Entry type

Theorem

Classification

msc 13F07

Related topic

FieldAdjunction

Related topic

PolynomialRingOverIntegralDomain

Related topic

PolynomialRingWhichIsPID

Defines

coprime

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