Suppose there are variables,
. A
monomial over a field
is a product of the form
|
(4.47) |
in which all of the exponents , , ,
are positive integers. The total degree of the
monomial is
.
A polynomial in variables
with
coefficients in
is a finite linear combination of monomials that
have coefficients in
. A polynomial can be expressed as
|
(4.48) |
in which is a monomial as shown in (4.47), and
is a coefficient. If
, then each is called a term. Note that the exponents may be
different for every term of . The total degree of
is the maximum total degree among the
monomials of the terms of . The set of all polynomials in
with coefficients in
is denoted by
.
Example 4..17 (Polynomials)
The definitions correspond exactly to our intuitive notion of a
polynomial. For example, suppose
. An example of a
polynomial in
is
|
(4.49) |
Note that
is a valid monomial; hence, any element of
may
appear alone as a term, such as the
in the polynomial
above. The total degree of (
4.49) is
due to the
second term. An equivalent polynomial may be written using nicer
variables. Using
,
, and
as variables yields
|
(4.50) |
which belongs to
.
The set
of polynomials is actually a group
with respect to addition; however, it is not a field. Even though
polynomials can be multiplied, some polynomials do not have a
multiplicative inverse. Therefore, the set
is
often referred to as a commutative ring of polynomials. A
commutative ring is a set with two operations for which every axiom
for fields is satisfied except the last one, which would require a
multiplicative inverse.
Steven M LaValle
2020-08-14