a=

i=’∞

where am , am’1 , . . . ∈ R. Thus, in a reversed formal Laurent series, we allow

an in¬nite number of terms involving negative powers of X, but only a ¬nite

number of terms involving positive powers of X.

The rules for addition and multiplication of reversed formal Laurent series

are just as one would expect: if

m m

i

bi Xi ,

a= ai X and b =

i=’∞ i=’∞

then

m

(ai + bi )Xi ,

a + b := (17.8)

i=’∞

and

2m m

ak bi’k Xi .

a · b := (17.9)

i=’∞ k=i’m

The ring of all reversed formal Laurent series is denoted R((X’1 )), and as

the notation suggests, the map that sends X to X’1 (and acts as the identity

on R) is an isomorphism of R((X)) with R((X’1 )).

Now, for any a = m ’1

i

i=’∞ ai X ∈ R((X )) with am = 0, let us de¬ne the

degree of a, denoted deg(a), to be the value m, and the leading coe¬-

cient of a, denoted lc(a), to be the value am . As for ordinary polynomials,

we de¬ne the degree of 0 to be ’∞, and the leading coe¬cient of 0 to be 0.

Note that if a happens to be a polynomial, then these de¬nitions of degree

and leading coe¬cient agree with that for ordinary polynomials.

382 More rings

Theorem 17.24. For a, b ∈ R((X’1 )), we have deg(ab) ¤ deg(a) + deg(b),

where equality holds unless both lc(a) and lc(b) are zero divisors. Fur-

thermore, if b = 0 and lc(b) is a unit, then b is a unit, and we have

deg(ab’1 ) = deg(a) ’ deg(b).

Proof. Exercise. 2

It is also natural to de¬ne a ¬‚oor function for reversed formal Laurent

series: for a ∈ R((X’1 )) with a = m i

i=’∞ ai X , we de¬ne

m

ai Xi ∈ R[X];

a :=

i=0

that is, we compute the ¬‚oor function by simply throwing away all terms

involving negative powers of X.

Now, let a, b ∈ R[X] with b = 0 and lc(b) a unit, and using the usual

division with remainder property for polynomials, write a = bq + r, where

q, r ∈ R[X] with deg(r) < deg(b). Let b’1 denote the multiplicative inverse of

b in R((X’1 )). It is not too hard to see that ab’1 = q; indeed, multiplying

the equation a = bq+r by b’1 , we obtain ab’1 = q+rb’1 , and deg(rb’1 ) < 0,

from which it follows that ab’1 = q.

Let F be a ¬eld. Since F ((X’1 )) is isomorphic to F ((X)), and the latter

is a ¬eld, it follows that F ((X’1 )) is a ¬eld. Now, F ((X’1 )) contains F [X]

as a subring, and hence contains (an isomorphic copy) of F (X). Just as

F (X) corresponds to the ¬eld of rational numbers, F ((X’1 )) corresponds to

the ¬eld real numbers. Indeed, we can think of real numbers as decimal

numbers with a ¬nite number of digits to the left of the decimal point

and an in¬nite number to the right, and reversed formal Laurent series

have a similar “syntactic” structure. In many ways, this syntactic similarity

between the real numbers and reversed formal Laurent series is more than

just super¬cial.

Exercise 17.16. Write down the rule for determining the multiplicative

inverse of an element of R((X’1 )) whose leading coe¬cient is a unit in R.

Exercise 17.17. Let F be a ¬eld of characteristic other than 2. Show that

a non-zero z ∈ F ((X’1 )) has a square-root in z ∈ F ((X’1 )) if and only if

deg(z) is even and lc(z) has a square-root in F .

Exercise 17.18. Let R be a ring, and let ± ∈ R. Show that the multiplica-

tive inverse of X ’ ± in R((X’1 )) is ∞ ±j’1 X’j .

j=1

17.8 Unique factorization domains (—) 383

Exercise 17.19. Let R be an arbitrary ring, let ±1 , . . . , ± ∈ R, and let

f := (X ’ ±1 )(X ’ ±2 ) · · · (X ’ ± ) ∈ R[X].

For j ≥ 0, de¬ne the “power sum”

j

sj := ±i .

i=1

Show that in the ring R((X’1 )), we have

∞

D(f ) 1

sj’1 X’j ,

= =

(X ’ ±i )

f

i=1 j=1

where D(f ) is the formal derivative of f .

Exercise 17.20. Continuing with the previous exercise, derive Newton™s

identities, which state that if f = X +f1 X ’1 +· · ·+f , with f1 , . . . , f ∈ R,

then

s1 + f1 = 0

s2 + f1 s1 + 2f2 = 0

s3 + f1 s2 + f2 s1 + 3f3 = 0

.

.

.

+ ··· + f

s + f1 s ’1 s1 + f =0

’1

+ ··· + f + f sj = 0 (j ≥ 1).

sj+ + f1 sj+ ’1 sj+1

’1

17.8 Unique factorization domains (—)

As we have seen, both the integers and the ring F [X] of polynomials over

a ¬eld enjoy a unique factorization property. These are special cases of a

more general phenomenon, which we explore here.

Throughout this section, D denotes an integral domain.

We call a, b ∈ D associate if a = ub for some u ∈ D— . Equivalently, a

and b are associate if and only if a | b and b | a. A non-zero element p ∈ D

is called irreducible if it is not a unit, and all divisors of p are associate to

1 or p. Equivalently, a non-zero, non-unit p ∈ D is irreducible if and only if

it cannot be expressed as p = ab where neither a nor b are units.

384 More rings

De¬nition 17.25. We call D a unique factorization domain (UFD)

if

(i) every non-zero element of D that is not a unit can be written as a

product of irreducibles in D, and

(ii) such a factorization into irreducibles is unique up to associates and

the order in which the factors appear.

Another way to state part (ii) of the above de¬nition is that if p1 · · · pr and

p1 · · · ps are two factorizations of some element as a product of irreducibles,

then r = s, and there exists a permutation π on the indices {1, . . . , r} such

that pi and pπ(i) are associate.

As we have seen, both Z and F [X] are UFDs. In both of those cases,

we chose to single out a distinguished irreducible element among all those

associate to any given irreducible: for Z, we always chose p to be positive,