.

2a

Exercise 9.15. Let R be a ring, let a ∈ R[X], with deg(a) = k ≥ 0, and let

± be an element of R.

(a) Show that there exists an integer m, with 0 ¤ m ¤ k, and a polyno-

mial q ∈ R[X], such that

a = (X ’ ±)m q and q(±) = 0R .

(b) Show that the values m and q in part (a) are uniquely determined

(by a and ±).

(c) Show that m > 0 if and only if ± is a root of a.

Let m± (a) denote the value m in the previous exercise; for completeness,

one can de¬ne m± (a) := ∞ if a is the zero polynomial. If m± (a) > 0, then

± is called a root of a of multiplicity m± (a); if m± (a) = 1, then ± is called

a simple root of a, and if m± (a) > 1, then ± is called a multiple root of

a.

The following exercise re¬nes Theorem 9.14, taking into account multi-

plicities.

Exercise 9.16. Let D be an integral domain, and let a ∈ D[X], with

deg(a) = k ≥ 0. Show that

m± (a) ¤ k.

±∈D

Exercise 9.17. Let D be an integral domain, let a, b ∈ D[X], and let ± ∈ D.

Show that m± (ab) = m± (a) + m± (b).

9.2 Polynomial rings 227

Exercise 9.18. Let R be a ring, let a ∈ R[X], with deg(a) = k ≥ 0, let

± ∈ R, and let m := m± (a). Show that if we evaluate a at X + ±, we have

k

bi Xi ,

a X+± =

i=m

where bm , . . . , bk ∈ R and bm = 0R .

9.2.4 Formal derivatives

i,

Let R be any ring, and let a ∈ R[X] be a polynomial. If a = i=0 ai X we

de¬ne the formal derivative of a as

iai Xi’1 .

D(a) :=

i=1

We stress that unlike the “analytical” notion of derivative from calculus,

which is de¬ned in terms of limits, this de¬nition is purely “symbolic.”

Nevertheless, some of the usual rules for derivatives still hold:

Theorem 9.17. Let R be a ring. For all a, b ∈ R[X] and c ∈ R, we have

(i) D(a + b) = D(a) + D(b);

(ii) D(ca) = cD(a);

(iii) D(ab) = D(a)b + aD(b).

Proof. Parts (i) and (ii) follow immediately by inspection, but part (iii)

requires some proof. First, note that part (iii) holds trivially if either a or b

are zero, so let us assume that neither are zero.

We ¬rst prove part (iii) for monomials, that is, polynomials of the form

i for non-zero c ∈ R and i ≥ 0. Suppose a = cXi and b = dXj . If

cX

i = 0, so a = c, then the result follows from part (ii) and the fact that

D(c) = 0; when j = 0, the result holds by a symmetric argument. So

assume that i > 0 and j > 0. Now, D(a) = icXi’1 and D(b) = jdXj’1 ,

and D(ab) = D(cdXi+j ) = (i + j)cdXi+j’1 . The result follows from a simple

calculation.

Having proved part (iii) for monomials, we now prove it in general on

induction on the total number of monomials appearing in a and b. If the

total number is 2, then both a and b are monomials, and we are in the base

case; otherwise, one of a and b must consist of at least two monomials, and

for concreteness, say it is b that has this property. So we can write b = b1 +b2 ,

where both b1 and b2 have fewer monomials than does b. Applying part (i)

228 Rings

and the induction hypothesis for part (iii), we have

D(ab) = D(ab1 + ab2 )

= D(ab1 ) + D(ab2 )

= D(a)b1 + aD(b1 ) + D(a)b2 + aD(b2 )

= D(a) · (b1 + b2 ) + a · (D(b1 ) + D(b2 ))

= D(a) · (b1 + b2 ) + a · D(b1 + b2 )

= D(a)b + aD(b). 2

Exercise 9.19. Let R be a ring, let a ∈ R[X], and let ± ∈ R be a root of

a. Show that ± is a multiple root of a if and only if ± is a root of D(a) (see

Exercise 9.15).

Exercise 9.20. Let R be a ring, let a ∈ R[X] with deg(a) = k ≥ 0, and let

± ∈ R. Show that if we evaluate a at X + ±, writing

k

bi Xi ,

a X+± =

i=0

with b0 , . . . , bk ∈ R, then we have

i! · bi = (Di (a))(±) for i = 0, . . . , k.

Exercise 9.21. Let F be a ¬eld such that every non-constant polynomial

a ∈ F [X] has a root ± ∈ F . (The ¬eld C is an example of such a ¬eld, an

important fact which we shall not be proving in this text.) Show that for

every positive integer r that is not a multiple of the characteristic of F , there

exists an element ζ ∈ F — of multiplicative order r, and that every element

in F — whose order divides r is a power of ζ.

9.2.5 Multi-variate polynomials

One can naturally generalize the notion of a polynomial in a single variable

to that of a polynomial in several variables. We discuss these ideas brie¬‚y

here”they will play only a minor role in the remainder of the text.

Consider the ring R[X] of polynomials over a ring R. If Y is another indeter-

minate, we can form the ring R[X][Y] of polynomials in Y whose coe¬cients

are themselves polynomials in X over the ring R. One may write R[X, Y]

instead of R[X][Y]. An element of R[X, Y] is called a bivariate polynomial.

9.2 Polynomial rings 229

Consider a typical element a ∈ R[X, Y], which may be written

k

aij Xi Yj .

a= (9.3)

j=0 i=0

Rearranging terms, this may also be written as

aij Xi Yj ,

a= (9.4)

0¤i¤k

0¤j¤

or as

k

aij Yj Xj .

a= (9.5)

i=0 j=0

If a is written as in (9.4), the terms aij Xi Yj with aij = 0R are called

monomials. The total degree of such a monomial aij Xi Yj is de¬ned to be

i + j, and if a is non-zero, then the total degree of a, denoted Deg(a), is

de¬ned to be the maximum total degree of any monomial appearing in (9.4).

We de¬ne the total degree of the zero polynomial to be ’∞. The reader

may verify that for any a, b ∈ R[X, Y], we have Deg(ab) ¤ Deg(a) + Deg(b),

while equality holds if R is an integral domain.

When a is written as in (9.5), one sees that we can naturally view a as

an element of R[Y][X], that is, as a polynomial in X whose coe¬cients are

polynomials in Y . From a strict, syntactic point of view, the rings R[Y][X]

and R[X][Y] are not the same, but there is no harm done in blurring this

distinction when convenient. We denote by degX (a) the degree of a, viewed

as a polynomial in X, and by degY (a) the degree of a, viewed as a polynomial

in Y. Analogously, one can formally di¬erentiate a with respect to either X

or Y, obtaining the “partial” derivatives DX (a) and DY (a).

Example 9.29. Let us illustrate, with a particular example, the three dif-