For nonzero
a
∈
R
[
T
], if
a
=
∑
k
i
=0
a
i
T
i
with
a
k
6
= 0
R
, we call
k
the
degree
of
a
, denoted
deg(
a
), and we call
a
k
the
leading coefficient
of
a
, denoted lc(
a
), and we call
a
0
the
constant
term
of
a
. If lc(
a
) = 1
R
, then
a
is called
monic
.
Note that if
a, b
∈
R
[
T
], both nonzero, and their leading coefficients are not both zero divisors,
then the product
ab
is nonzero and deg(
ab
) = deg(
a
) + deg(
b
). However, if the leading coefficients
of
a
and
b
are both zero divisors, then we could get some “collapsing”: we could have
ab
= 0
R
, or
ab
6
= 0
R
but deg(
ab
)
<
deg(
a
) + deg(
b
).
For the zero polynomial, we establish the following conventions:
its leading coefficient and
constant term are defined to be 0
R
, and its degree is defined to be “
∞
”, where it is understood
that for all integers
x
∈
Z
,
∞
< x
, and (
∞
) +
x
=
x
+ (
∞
) =
∞
, and (
∞
) + (
∞
) =
∞
.
This notion of “negative infinity” should not be construed as a useful algebraic notion — it is
simply a convenience of notation; for example, it allows us to succinctly state that
for all
a, b
∈
R
[
T
], deg(
ab
)
≤
deg(
a
) + deg(
b
), with equality holding if the leading
coefficients of
a
and
b
are not both zero divisors.
Theorem 5.5
Let
D
be an integral domain. Then
1. for all
a, b
∈
D
[
T
]
,
deg(
ab
) = deg(
a
) + deg(
b
)
;
2.
D
[
T
]
is an integral domain;
3.
(
D
[
T
])
*
=
D
*
.
Proof.
Exercise.
2
5.2.3
Division with remainder
An extremely important property of polynomials is a division with remainder property, analogous
to that for the integers:
Theorem 5.6 (Division with Remainder Property)
Let
R
be a ring.
For
a, b
∈
R
[
T
]
with
lc(
b
)
∈
R
*
, there exist unique
q, r
∈
R
[
T
]
such that
a
=
bq
+
r
and
deg(
r
)
<
deg(
b
)
.
Proof.
Consider the set
S
of polynomials of the form
a

xb
with
x
∈
R
[
T
]. Let
r
=
a

qb
be an
element of
S
of minimum degree. We must have deg(
r
)
<
deg(
b
), since otherwise, we would have
r
0
:=
r

(lc(
r
)lc(
b
)

1
T
deg(
r
)

deg(
b
)
)
·
b
∈
S
, and deg(
r
0
)
<
deg(
r
), contradicting the minimality of
deg(
r
).
That proves the existence of
r
and
q
. For uniqueness, suppose that
a
=
bq
+
r
and
a
=
bq
0
+
r
0
,
where deg(
r
)
<
deg(
b
) and deg(
r
0
)
<
deg(
b
). This implies
r
0

r
=
b
(
q

q
0
)
.
However, if
q
6
=
q
0
,
then
deg(
b
)
>
deg(
r
0

r
) = deg(
b
(
q

q
0
)) = deg(
b
) + deg(
q

q
0
)
≥
deg(
b
)
,
which is impossible. Therefore, we must have
q
=
q
0
, and hence
r
=
r
0
.
2
If
a
=
bq
+
r
as in the above theorem, we define
a
rem
b
:=
r
.