an algebraic extension of E, then K is an algebraic extension of F .

Exercise 17.13. Let E be an extension of F . Show that the set of all

elements in E that are algebraic over F is a sub¬eld of E containing F .

We close this section with a discussion of a splitting ¬eld ” a ¬nite

extension of the coe¬cient ¬eld in which a given polynomial splits completely

into linear factors. As the next theorem shows, splitting ¬elds always exist.

Theorem 17.19. Let F be a ¬eld, and f ∈ F [X] a monic polynomial of

degree . Then there exists a ¬nite extension K of F in which f factors as

f = (X ’ ±1 )(X ’ ±2 ) · · · (X ’ ± ),

with ±1 , . . . , ± ∈ K.

Proof. We prove the existence of K by induction on the degree of f . If

= 0, then the theorem is trivially true. Otherwise, let g be an irreducible

factor of f , and set E := F [X]/(g), so that ± := [X]g is a root of g, and hence

of f , in E. So over the extension ¬eld E, f factors as

f = (X ’ ±)h,

where h ∈ E[X] is a polynomial of degree ’ 1. Applying the induction

hypothesis, there exists a ¬nite extension K of E such that h splits into

linear factors over K. Thus, over K, f splits into linear factors, and by

Exercise 17.11, K is a ¬nite extension of F . 2

17.7 Formal power series and Laurent series

We discuss generalizations of polynomials that allow an in¬nite number of

non-zero coe¬cients. Although we are mainly interested in the case where

the coe¬cients come from a ¬eld F , we develop the basic theory for general

rings R.

17.7 Formal power series and Laurent series 379

17.7.1 Formal power series

The ring R[[X]] of formal power series over R consists of all formal ex-

pressions of the form

a = a0 + a1 X + a2 X2 + · · · ,

where a0 , a1 , a2 , . . . ∈ R. Unlike ordinary polynomials, we allow an in¬nite

number of non-zero coe¬cients. We may write such a formal power series

as

∞

ai Xi .

a=

i=0

The rules for addition and multiplication of formal power series are exactly

the same as for polynomials. Indeed, the formulas (9.1) and (9.2) in §9.2

for addition and multiplication may be applied directly” all of the relevant

sums are ¬nite, and so everything is well de¬ned.

We shall not attempt to interpret a formal power series as a function, and

therefore, “convergence” issues shall simply not arise.

Clearly, R[[X]] contains R[X] as a subring. Let us consider the group of

units of R[[X]].

∞

∈ R[[X]]. Then a ∈ (R[[X]])— if and only

i

Theorem 17.20. Let a = i=0 ai X

if a0 ∈ R— .

Proof. If a0 is not a unit, then it is clear that a is not a unit, since the

constant term of a product formal power series is equal to the product of

the constant terms.

Conversely, if a0 is a unit, we show how to de¬ne the coe¬cients of the

inverse b = ∞ bi Xi of a. Let ab = c = ∞ ci Xi . We want c = 1, meaning

i=0 i=0

that c0 = 1 and ci = 0 for all i > 0. Now, c0 = a0 b0 , so we set b0 := a’1 . 0

’1

Next, we have c1 = a0 b1 + a1 b0 , so we set b1 := ’a1 b0 · a0 . Next, we have

c2 = a0 b2 + a1 b1 + a2 b0 , so we set b2 := ’(a1 b1 + a2 b0 ) · a’1 . Continuing in

0

this way, we see that if we de¬ne bi := ’(a1 bi’1 + · · · + ai b0 ) · a’1 for i ≥ 1,

0

then ab = 1. 2

Example 17.11. In the ring R[[X]], the multiplicative inverse of 1 ’ X is

∞

i=0 X . 2

i

Exercise 17.14. For a ¬eld F , show that any non-zero ideal of F [[X]] is of

the form (Xm ) for some uniquely determined integer m ≥ 0.

380 More rings

17.7.2 Formal Laurent series

One may generalize formal power series to allow a ¬nite number of negative

powers of X. The ring R((X)) of formal Laurent series over R consists of

all formal expressions of the form

a = am Xm + am+1 Xm+1 + · · · ,

where m is allowed to be any integer (possibly negative), and am , am+1 , . . . ∈

R. Thus, elements of R((X)) may have an in¬nite number of terms involving

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

powers of X. We may write such a formal Laurent series as

∞

ai Xi .

a=

i=m

The rules for addition and multiplication of formal Laurent series are just

as one would expect: if

∞ ∞

i

bi Xi ,

a= ai X and b =

i=m i=m

then

∞

(ai + bi )Xi ,

a + b := (17.6)

i=m

and

∞ i’m

ak bi’k Xi .

a · b := (17.7)

i=2m k=m

We leave it to the reader to verify that R((X)) is a ring containing R[[X]].

Theorem 17.21. If D is an integral domain, then D((X)) is an integral

domain.

Proof. Let a = ∞ ai Xi and b = ∞ bi Xi , where am = 0 and bn = 0.

i=m i=n

Then ab = ∞ ci , where cm+n = am bn = 0. 2

i=m+n

∞ i

Theorem 17.22. Let a ∈ R((X)), and suppose that a = 0 and a = i=m ai X

with am ∈ R— . Then a has a multiplicative inverse in R((X)).

Proof. We can write a = Xm b, where b is a formal power series whose constant

term is a unit, and hence there is a formal power series c such that bc = 1.

Thus, X’m c is the multiplicative inverse of a in R((X)). 2

As an immediate corollary, we have:

17.7 Formal power series and Laurent series 381

Theorem 17.23. If F is a ¬eld, then F ((X)) is a ¬eld.

Exercise 17.15. Show that for a ¬eld F , F ((X)) is the ¬eld of fractions of

F [[X]]; that is, there is no proper sub¬eld of F ((X)) that contains F [[X]].

17.7.3 Reversed formal Laurent series

While formal Laurent series are useful in some situations, in many others,

it is more useful and natural to consider reversed formal Laurent series

over R. These are formal expressions of the form

m