Graphics Reference
In-Depth Information
can then be expressed in the more usual form as
•
Â
()
=
2
i
ffX a aXaX
=
+
+
+ ◊◊◊=
aX
i
.
0
1
2
i
=
0
In the case of a polynomial of degree n, one writes f in the form
n
Â
2
n
i
()
=
ffX a aXaX
=
+
+
+ ◊◊◊+
aX
=
aX
.
(B.2)
0
1
2
n
i
i
=
0
Definition.
The term a
n
X
n
in expression (B.2) is called the
leading term
of f and a
n
is called the
leading coefficient
of f and are denoted by lt(f) and lc(f), respectively.
B.7.1. Theorem.
If R[[X]] is the set of formal power series over R, then (R[[X]],+,·)
is a ring. If R is commutative, then so is R[[X]]. If R is an integral domain, then so is
R[[X]]. If R[X] Ã R[[X]] is the set of polynomials over R, then (R[X],+,·) is a subring
of R[[X]].
Proof.
Easy.
Definition.
R[[X]] is called the
formal power series ring over R
(in the indeterminate
X). R[X] is called the
polynomial ring over R
(in the indeterminate X).
Formal power series are the algebraic analog of power series in calculus. The
difference is that we do not worry about convergence here and there is no need to
define a topology for the ring.
B.7.2. Theorem.
If R is a subring of a ring Q and if u is an element of Q that is
transcendental over R, then R[X] is isomorphic to R[u].
Proof.
The theorem follows from a
universal factorization property
satisfied by R[X]
with respect to ring homomorphisms, namely,
Let R¢ be a subring of Q¢ and let u¢ Œ Q¢. Any ring homomorphism h : R Æ R¢
extends to a unique ring homomorphism H : R[X] Æ R¢[u¢], which maps X to u¢.
B.7.3. Theorem.
If R is a UFD, then both R[[X]] and R[X] are UFDs. The only
irreducible element in R[[X]], up to unit, is X.
Proof.
For R[X] see [Dean66], for example. For R[[X]] see [Seid68].
Let R be a ring and let X
1
, X
2
,...be symbols (or
indeterminates
). Define rings R
((n))
and R
(n)
recursively by
(
(
)
)
0
RR
R
=
=
(
(
)
)
(
(
)
)
n
n
-
1
[
[
]
]
R
X
,
for
n
>
0
,
n
and
