Polynomials
This chaptser provides a continuation of Chapter II, §3. We prove standard properties of polynomials. Most readers will be acquainted with some of these properties, especially at the beginning for polynomials in one variable. However, one of our purposes
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IV
Polynomials
This chapter provides a continuation of Chapter II, §3. We prove standard properties of polynomials. Most readers will be acquainted with some of these properties, especially at the beginning for polynomials in one variable. However, one of our purposes is to show that some of these properties also hold over a commutative ring when properly formulated. The Gauss lemma and the reduction criterion for irreducibility will show the importance of working over rings. Chapter IX will give examples of the importance of working over the integers Z themselves to get universal relations. It happens that certain statements of algebra are universally true. To prove them, one proves them first for elements of a polynomial ring over Z, and then one obtains the statement in arbitrary fields (or commutative rings as the case may be) by specialization. The Cayley-Hamilton theorem of Chapter XV, for instance, can be proved in that way. The last section on power series shows that the basic properties of polynomial rings can be formulated so as to hold for power series rings. I conclude this section with several examples showing the importance of power series in various parts of mathematics.
§1.
BASIC PROPERTIES FOR POLYNOMIALS IN ONE VARIABLE
We start with the Euclidean algorithm.
Theorem 1.1. Let A be a commutative ring, let f, g E A[X] be polynomials in one variable, of degrees ~ 0, and assume that the leading 173 S. Lang, Algebra © Springer Science+Business Media LLC 2002
174
IV, §1
POLYNOMIALS
coefficient of 9 is a unit in A. Then there exist unique polynomials q, r E A [X] such that
f = gq + r and deg r < deg g. Proof
Write
f(X)
= anX n +
g(X) = bdx d +
+ ao, + bo,
where n = deg J, d = deg 9 so that an, bd =J 0 and bd is a unit in A. We use induction on n. If n = 0, and deg 9 > deg J, we let q = 0, r = [. If deg 9 = deg f = 0, then we let r = 0 and q = an bi 1 • Assume the theorem proved for polynomials of degree < n (with n > 0). We may assume deg 9 ;:£ deg f (otherwise, take q = 0 and r = f). Then
where f1 (X) has degree < n. By induction, we can find q l ' r such that
and deg r < deg g. Then we let
to conclude the proof of existence for q, r. As for uniqueness, suppose that
with deg r1 < deg 9 and deg rz < deg g. Subtracting yields
Since the leading coefficient of 9 is assumed to be a unit, we have deg(ql - qz)g = deg(q1 - qz)
+ deg g.
Since deg(r z - rd < deg g, this relation can hold only if q1 - qz = 0, i.e. q1 = qz, and hence finally r 1 = rz as was to be shown.
Theorem 1.2. Let k be a field. Then the polynomial ring in one variable k[X] is principal.
IV, §1
BASIC PROPERTIES FOR POLYNOMIALS IN ONE VARIABLE
175
Proof. Let a be an ideal of k[X] , and assume a =I- O. Let 9 be an element of a of smallest degree ~ O. Let f be any element of a such that f =I- O. By the Euclidean algorithm we can find q, r E k[X] such that
f
= qg
+r
and deg r < deg g. But r = f - qg, whence r is in a. Since g had minimal degree ~ 0 it follows that r = 0, hence that a consists of all polynomials qg
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