polynomial ring

polynomial ring

1

Polynomial rings in one variable

Let

R

be a ring. The

polynomial ring

over

R

in one

variable

X

is the set

R

⁢

[

X

]

of all

sequences

in

R

with only finitely many nonzero terms. If

(

a

0

,

a

1

,

a

2

,

a

3

,

…

)

is an

element

in

R

⁢

[

X

]

, with

a

n

=

0

for all

n

>

N

, then we usually write this element as

∑

n

=

0

N

a

n

⁢

X

n

=

a

0

+

a

1

⁢

X

+

a

2

⁢

X

2

+

a

3

⁢

X

3

+

⋯

+

a

N

⁢

X

N

.

Elements of

R

⁢

[

X

]

are called

polynomials

in the

indeterminate

X

with

coefficients

in

R

. The ring elements

a

0

,

…

,

a

N

are called

coefficients

of the polynomial, and the

degree

of a polynomial is the largest

natural number

N

for which

a

N

≠

0

, if such an

N

exists. When a polynomial has all of its coefficients equal to

0

, its degree is usually considered to be undefined, although some people adopt the convention that its degree is

-

∞

.

A

monomial

is a polynomial with exactly one nonzero coefficient. Similarly, a

binomial

is a polynomial with exactly two nonzero coefficients, and a

trinomial

is a polynomial with exactly three nonzero coefficients.

Addition

and multiplication of polynomials is defined by

∑

n

=

0

N

a

n

⁢

X

n

+

∑

n

=

0

N

b

n

⁢

X

n

=

∑

n

=

0

N

(

a

n

+

b

n

)

⁢

X

n

(1)

∑

n

=

0

N

a

n

⁢

X

n

⋅

∑

n

=

0

N

b

n

⁢

X

n

=

∑

n

=

0

2

⁢

N

(

∑

k

=

0

n

a

k

⁢

b

n

-

k

)

⁢

X

n

(2)

R

⁢

[

X

]

is a

ℤ

–

graded ring

under these

operations

, with the monomials of degree exactly

n

comprising the

n

th

graded

component

of

R

⁢

[

X

]

. The

zero element

of

R

⁢

[

X

]

is the polynomial whose coefficients are all

0

, and when

R

has a

multiplicative identity

1

, the polynomial whose coefficients are all

0

except for

a

0

=

1

is a multiplicative identity for the polynomial ring

R

⁢

[

X

]

.

2

Polynomial rings in finitely many variables

The

polynomial ring

over

R

in two variables

X

,

Y

is defined to be

R

⁢

[

X

,

Y

]

:=

R

⁢

[

X

]

⁢

[

Y

]

≅

R

⁢

[

Y

]

⁢

[

X

]

. Elements of

R

⁢

[

X

,

Y

]

are called

polynomials

in the indeterminates

X

and

Y

with coefficients in

R

. A

monomial

in

R

⁢

[

X

,

Y

]

is a polynomial which is simultaneously a monomial in both

X

and

Y

, when considered as a polynomial in

X

with coefficients in

R

⁢

[

Y

]

(or as a polynomial in

Y

with coefficients in

R

⁢

[

X

]

). The

degree

of a monomial in

R

⁢

[

X

,

Y

]

is the sum of its

individual

degrees in the respective indeterminates

X

and

Y

(in

R

⁢

[

Y

]

⁢

[

X

]

and

R

⁢

[

X

]

⁢

[

Y

]

), and the degree of a polynomial in

R

⁢

[

X

,

Y

]

is the

supremum

of the degrees of its monomial summands, if it has any.

In three variables, we have

R

⁢

[

X

,

Y

,

Z

]

:=

R

⁢

[

X

,

Y

]

⁢

[

Z

]

=

R

⁢

[

X

]

⁢

[

Y

]

⁢

[

Z

]

≅

R

⁢

[

X

]

⁢

[

Z

]

⁢

[

Y

]

≅

⋯

, and in any finite number of variables, we have inductively

R

⁢

[

X

1

,

X

2

,

…

,

X

n

]

:=

R

⁢

[

X

1

,

…

,

X

n

-

1

]

⁢

[

X

n

]

=

R

⁢

[

X

1

]

⁢

[

X

2

]

⁢

⋯

⁢

[

X

n

]

, with monomials and degrees defined in

analogy

to the two variable case. In any number of variables, a polynomial ring is a graded ring with

n

th

graded component equal to the

R

-module generated by the monomials of degree

n

.

3

Polynomial rings in arbitrarily many variables

For any

nonempty set

M

, let

E

⁢

(

M

)

denote the set of all finite subsets of

M

. For each element

A

=

{

a

1

,

…

,

a

n

}

of

E

⁢

(

M

)

, set

R

⁢

[

A

]

:=

R

⁢

[

a

1

,

…

,

a

n

]

. Any two elements

A

,

B

∈

E

⁢

(

M

)

satisfying

A

⊂

B

give rise to the relationship

R

⁢

[

A

]

⊂

R

⁢

[

B

]

if we consider

R

⁢

[

A

]

to be embedded in

R

⁢

[

B

]

in the obvious way. The union of the rings

{

R

⁢

[

A

]

:

A

∈

E

⁢

(

M

)

}

(or, more formally, the

categorical

direct limit

of the

direct system

of rings

{

R

⁢

[

A

]

:

A

∈

E

⁢

(

M

)

}

) is defined to be the ring

R

⁢

[

M

]

.

Title

polynomial ring

Canonical name

PolynomialRing

Date of creation

2013-03-22 11:52:27

Last modified on

2013-03-22 11:52:27

Owner

djao (24)

Last modified by

djao (24)

Numerical id

10

Author

djao (24)

Entry type

Definition

Classification

msc 11C08

Classification

msc 12E05

Classification

msc 13P05

Classification

msc 17B66

Classification

msc 16W10

Classification

msc 70G65

Classification

msc 17B45

Related topic

AlgebraicGeometry

Related topic

RationalFunction

Defines

polynomial

Defines

monomial

Defines

binomial

Defines

trinomial

Defines

degree

Generated on Fri Feb 9 15:30:59 2018 by

LaTeXML