Graphics Reference
In-Depth Information
It is easy to show that the intersection of subrings is a subring, so that A[S] is a
subring. Furthermore, if A is a subring of a commutative ring R with unity and if u
is any element of R, then it is easy to show that
{
}
[]
=
2
2
n
A u
a
+
a u
+
a u
+◊◊◊+
a u
a
Œ
R
and
n
≥
0.
01
n
i
This justifies calling A[u] a polynomial ring. We would like to define the polynomial
ring R[X] in an “indeterminate” (or “variable”) X, where “indeterminate” basically
means that X is transcendental over R in the following sense:
Definition.
Let A be a subring of a commutative ring R with unity and let u ΠR.
We say that u is
transcendental
over A if
2
n
aauau
+
+
+ ◊◊◊+
au
n
=
0
0
1
2
with a
i
ΠR implies that a
i
= 0 for all i. If u is not transcendental, then u is said to be
algebraic
over A.
The definition of R[X] boils down to constructing an object with the desired
properties. Because polynomials are a special case of formal power series, we shall
deal with both simultaneously so as not to have to duplicate basically the same defi-
nitions later on.
Definition.
Let R be a ring. A
formal power series
over R is an infinite sequence
(a
0
,a
1
,a
2
, . . .) where the a
i
are elements of R. If all but a finite number of the a
i
are
zero, then the sequence is called a
polynomial over R
. The power series for which all
the a
i
are zero is called the
zero power series
or
zero polynomial
and will be denoted
by 0. In the case of a nonzero polynomial, the largest index i for which a
i
is nonzero
is called the
degree
of the polynomial. The zero polynomial is said to have degree 0.
Let f = (a
0
,a
1
,a
2
, . . .) and g = (b
0
,b
1
,b
2
, . . .) be formal power series over a ring R.
Define an addition + and a multiplication · of formal power series as follows:
f
+
gababab
=
(
+
,
+
,
+
,...
)
0
0
1
1
2
2
and
k
Â
◊=
(
)
f
g
c
,
c
,
c
,... ,
where
c
=
a b
.
012
k
i
k
-
i
i
=
0
Let X be some symbol (or
indeterminate
). We shall identify the polynomial
( , ,..., , , ,...)
00
0
a
0
144
i
-
1
with the formal expression “aX
i
.” The terms “aX
0
” and “aX
1
” will be abbreviated to
“a” and “aX,” respectively. With this identification a formal power series
=
(
)
f
aaa
,
,
,...
012
