A138
A.2
Polynomial Algebra over Fields
A.2.1
Polynomial rings over fields
We have introduced fields in order to give arithmetic structure to our alphabet
F
.
Our next wish is then to give arithmetic structure to words formed from
our alphabet.
Additive structure has been provided by considering words as
members of the vector space
F
n
of
n
tuples from
F
for appropriate
n
. Scalar
multiplication in
F
n
does not however provide a comprehensive multiplication
for words and vectors. In order to construct a workable definition of multipli
cation for words, we introduce polynomials over
F
.
Let
F
be a field, and let
x
be a symbol not one of those of
F
, an
indetermi
nate
. To any
n
tuple
indeterminate
(
a
0
, a
1
, a
2
, . . . , a
n

1
)
of members of
F
we associate the
polynomial
in
x
:
polynomial
a
0
x
0
+
a
1
x
1
+
a
2
x
2
+
· · ·
+
a
n

1
x
n

1
.
In keeping with common notation we usually write
a
0
for
a
0
x
0
and
a
1
x
for
a
1
x
1
.
Also we write 0
·
x
i
= 0 and 1
·
x
i
=
x
i
. We sometimes use summation notation
for polynomials:
d
X
i
=0
a
i
x
i
=
a
0
x
0
+
a
1
x
1
+
a
2
x
2
+
· · ·
+
a
d
x
d
.
We next define
F
[
x
] as the set of all polynomials in
x
over
F
:
F
[
x
]
F
[
x
] =
{
∞
X
i
=0
a
i
x
i

a
i
∈
F, a
i
= 0 for all but a finite number of
i
}
.
Polynomials are added in the usual manner:
∞
X
i
=0
a
i
x
i
+
∞
X
i
=0
b
i
x
i
=
∞
X
i
=0
(
a
i
+
b
i
)
x
i
.
This addition is compatible with vector addition of
n
tuples in the sense that
if the vector
a
of
F
n
is associated with the polynomial
a
(
x
) and the vector
b
is associated with the polynomial
b
(
x
), then the vector
a
+
b
is associated with
the polynomial
a
(
x
) +
b
(
x
).
Polynomial multiplication is also familiar:
∞
X
i
=0
a
i
x
i
·
∞
X
j
=0
b
j
x
j
=
∞
X
k
=0
c
k
x
k
,
where the coefficient
c
k
is given by convolution:
c
k
=
∑
i
+
j
=
k
a
i
b
j
. This multi
plication is the inevitable consequence of the distributive law provided we require
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A.2.
POLYNOMIAL ALGEBRA OVER FIELDS
A139
that
ax
i
·
bx
j
= (
ab
)
x
i
+
j
always. (As usual we shall omit the
·
in multiplication
when convenient.)
The set
F
[
x
] equipped with the operations + and
·
is the
polynomial ring
in
polynomial ring
x
over the field
F
.
F
is the field of
coefficients
of
F
[
x
].
coefficients
Polynomial rings over fields have many of the properties enjoyed by fields.
F
[
x
] is closed and distributive nearly by definition. Commutativity and additive
associativity for
F
[
x
] are easy consequences of the same properties for
F
, and
multiplicative associativity is only slightly harder to check. The constant poly
nomials 0
x
0
= 0 and 1
x
0
= 1 serve respectively as additive and multiplicative
identities. The polynomial
ax
0
=
a
, for
a
∈
F
, is usually referred to as a
con
stant polynomial
or a
scalar polynomial
. Indeed if we define scalar multiplication
constant polynomial
scalar polynomial
by
α
∈
F
as multiplication by the scalar polynomial
α
(=
αx
0
), then
F
[
x
] with
polynomial addition and this scalar multiplication is a vector space over
F
, a
basis for which is the subset
{
1
, x, x
2
, x
3
, . . . , x
i
, . . .
}
. (Note that (

1)
·
a
(
x
) is
the additive inverse of the polynomial
a
(
x
).)
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 Spring '11
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 A.2. POLYNOMIAL ALGEBRA

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