Basic Algebra - Computer Graphics and Geometric Modeling - page 859

B.11.13. Lemma. If f Πk[[X]] has a nonzero constant term (equivalently, ord(f) =
0), then there exists a g Πk[[X]] with fg = 1.

Proof. Simply write out the series expansions for f and g and use the equation
fg = 1 to solve for the series coefficients of g in terms of those of f.

B.11.14. Theorem. The units of k[[X]] are the elements f with ord(f) = 0.

Proof. Lemma B.11.13 proved half of the theorem. One still needs to show that every
unit has a nonzero constant term. This is left as an easy exercise for the reader.

B.11.15. Theorem. Every nonzero element f in k((X)) can be written uniquely in

the form

1

( )

2

f = aaXaX + + + ... ,

0 1 2

n

X

where a i Œ k and a 0 π 0.

Proof. See [Walk50].

Theorem B.11.15 leads to a definition of order that extends the one for formal
power series.

Definition. The integer n in Theorem B.11.15 is called the order of f and is denoted

by ord(f).

g is a homomor-
phism of R[[X]] into itself. We can strengthen this in the case where R is field.

Theorem B.7.5 can be interpreted as saying that the map f Æ f

B.11.16. Theorem. If f, g Œ k[[X]], where ord(g) = 1, then the map f Æ f g is an

order-preserving isomorphism of k[[X]], that is,

(1) ord(f) = ord(f g).

(2) There is a g¢Œk[[X]] with ord(g¢) = 1, so that f = (f

g) g¢ for all f Œ k[[X]].

Proof. See [Walk50].

B.12 Algebras

Definition. A ring A with unity that is a vector space over a field k and that satisfies

() = () = () u ab ua b a ub , for all u Πk a b , , ΠA ,

is called an algebra over k . If A is also a skew ring, then A is called a division algebra .