polynomial functions vs polynomials

polynomial functions vs polynomials

Let

k

be a field. Recall that a

function

f

:

k

→

k

is called

polynomial function

, iff there are

a

0

,

…

,

a

n

∈

k

such that

f

⁢

(

x

)

=

a

0

+

a

1

⁢

x

+

a

2

⁢

x

2

+

⋯

+

a

n

⁢

x

n

for any

x

∈

k

.

The ring of all polynomial functions (together with obvious

addition

and multiplication) we denote by

k

⁢

{

x

}

. Also denote by

k

⁢

[

x

]

the ring of polynomials (see this entry (

http://planetmath.org/PolynomialRing

) for details).

There is a canonical function

T

:

k

⁢

[

x

]

→

k

⁢

{

x

}

such that for any

polynomial

W

=

∑

i

=

1

n

a

i

⋅

x

i

we have that

T

⁢

(

W

)

is a polynomial function given by

T

⁢

(

W

)

⁢

(

x

)

=

∑

i

=

1

n

a

i

⋅

x

i

.

(Although we use the same notation for polynomials and polynomial functions these

concepts

are not the same). This function is called the

evaluation map

. As a simple exercise we leave the following to the reader:

Proposition

1.

The evaluation map

T

is a

ring homomorphisms

which is ,,onto”.

The question is: when

T

is ,,1-1”?

Proposition 2.

T

is ,,1-1” if and only if

k

is an

infinite

field.

Proof.

,,

⇒

” Assume that

k

=

{

a

1

,

…

,

a

n

}

is a

finite field

. Put

W

=

(

x

-

a

1

)

⁢

⋯

⁢

(

x

-

a

n

)

.

Then for any

x

∈

k

we have that

x

=

a

i

for some

i

and

T

⁢

(

W

)

⁢

(

x

)

=

(

x

-

a

1

)

⁢

⋯

⁢

(

x

-

a

n

)

=

(

a

i

-

a

1

)

⁢

⋯

⁢

(

a

i

-

a

i

)

⁢

⋯

⁢

(

a

i

-

a

n

)

=

0

which shows that

W

∈

Ker

⁢

T

although

W

is nonzero. Thus

T

is not ,,1-1”.

,,

⇐

” Assume, that

W

=

∑

i

=

1

n

a

i

⋅

x

i

is a polynomial with positive degree, i.e.

n

⩾

1

and

a

n

≠

0

such that

T

⁢

(

W

)

is a zero function. It follows from the Bezout’s

theorem

that

W

has at most

n

roots (in fact this is true over any

integral domain

). Thus since

k

is an infinite field, then there exists

a

∈

k

which is not a root of

W

. In particular

T

⁢

(

W

)

⁢

(

a

)

≠

0

.

Contradiction

, since

T

⁢

(

W

)

is a zero function. Thus

T

is ,,1-1”, which

completes

the proof.

□

This shows that the evaluation map

T

is an

isomorphism

only when

k

is infinite. So the interesting question is what is a kernel of

T

, when

k

is a finite field?

Proposition 3.

Assume that

k

=

{

a

1

,

…

,

a

n

}

is a finite field and

W

=

(

x

-

a

1

)

⁢

⋯

⁢

(

x

-

a

n

)

.

Then

T

⁢

(

W

)

=

0

and if

T

⁢

(

U

)

=

0

for some polynomial

U

, then

W

divides

U

. In particular

Ker

⁢

T

=

(

W

)

.

Proof.

In the proof of proposition 2 we’ve shown that

T

⁢

(

W

)

=

0

. Now if

T

⁢

(

U

)

=

0

, then every

a

i

is a root of

U

. It follows from the Bezout’s theorem that

(

x

-

a

i

)

must divide

U

for any

i

. In particular

W

divides

U

. This (together with the fact that

T

⁢

(

W

)

=

0

) shows that the ideal

Ker

⁢

T

is generated by

W

.

□

.

Corollary 4.

If

k

is a finite field of order

q

>

1

, then

k

⁢

{

x

}

has exactly

q

q

elements

.

Proof.

Let

k

=

{

a

1

,

…

,

a

q

}

and

W

=

(

x

-

a

1

)

⁢

⋯

⁢

(

x

-

a

q

)

.

By propositions 1 and 3 (and due to

First Isomorphism Theorem

for rings) we have that

k

⁢

{

x

}

≃

k

⁢

[

x

]

/

(

W

)

.

But the degree of

W

is equal to

q

. It follows that dimension of

k

⁢

[

x

]

/

(

W

)

(as a

vector space

over

k

) is equal to

dim

k

⁢

k

⁢

[

x

]

/

(

W

)

=

q

.

Thus

k

⁢

{

x

}

is isomorphic to

q

copies of

k

as a vector space

k

⁢

{

x

}

≃

k

×

⋯

×

k

.

This completes the proof, since

k

has

q

elements.

□

Remark.

Also all of this hold, if we replace

k

with an integral domain (we can always pass to its

field of fractions

). However this is not really interesting, since finite integral domains are exactly fields (Wedderburn’s little theorem).

Title

polynomial functions vs polynomials

Canonical name

PolynomialFunctionsVsPolynomials

Date of creation

2013-03-22 19:18:03

Last modified on

2013-03-22 19:18:03

Owner

joking (16130)

Last modified by

joking (16130)

Numerical id

4

Author

joking (16130)

Entry type

Theorem

Classification

msc 13A99

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