9.3. THE DERIVATIVE AND MULTIPLE ROOTS
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9.3. THE DERIVATIVE AND MULTIPLE ROOTS
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Corollary 9.2.5. Let p.x/ 2 KŒx, and suppose L and Q L are two splitting fields for p.x/. Then there is an isomorphism W L ! Q
L such that .k/ D k for all k 2 K.
Proof. Take K D Q K and D id in the proposition.
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Exercises 9.2 9.2.1. For any polynomial f .x/ 2 KŒx there is an extension field L of K such that f .x/ factors into linear factors in LŒx. Give a proof by induction on the degree of f .
9.2.2. Verify the following statements: The rational polynomial f .x/ D
x6 3 is irreducible over Q. It factors over Q.31=6/ as
.x
31=6/.x C 31=6/.x2
31=6x C 31=3/.x2 C 31=6x C 31=3/: If ! D ei=3, then the irreducible factorization of f .x/ over Q.31=6; !/
is
.x
31=6/.x C 31=6/.x !31=6/.x
!231=6/.x
!431=6/.x
!531=6/:
9.2.3.
(a) Show that if K is a finite field, then KŒx has irreducible elements of arbitrarily large degree. Hint: Use the existence of infinitely many irreducibles in KŒx, Proposition 1.8.9.
(b) Show that if K is a finite field, then K admits field extensions of arbitrarily large finite degree.
(c) Show that a finite–dimensional extension of a finite field is a finite field.
9.3. The Derivative and Multiple Roots
In this section, we examine, by means of exercises, multiple roots of polynomials and their relation to the formal derivative.
We say that a polynomial f .x/ 2 KŒx has a root a with multiplicity m in an extension field L if f .x/ D .x a/mg.x/ in LŒx, and g.a/ ¤ 0.
A root of multiplicity greater than 1 is called a multiple root. A root of multiplicity 1 is called a simple root.
The formal derivative in KŒx is defined by the usual rule from calcu lus: We define D.xn/ D nxn 1 and extend linearly. Thus, D.P knxn/ D
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9. FIELD EXTENSIONS – SECOND LOOK P nknxn 1. The formal derivative satisfies the usual rules for differentiation:
9.3.1. Show that D.f .x/Cg.x// D Df .x/CDg.x/ and D.f .x/g.x// D D.f .x//g.x/ C f .x/D.g.x//.
9.3.2.
(a) Suppose that the field K is of characteristic zero. Show that Df .x/ D 0 if, and only if, f .x/ is a constant polynomial.
(b) Suppose that the field has characteristic p. Show that Df .x/ D 0 if, and only if, there is a polynomial g.x/ such that f .x/ D g.xp/.
9.3.3. Suppose f .x/ 2 KŒx, L is an extension field of K, and f .x/ factors as f .x/ D .x a/g.x/ in LŒx. Show that the following are equivalent: (a) a is a multiple root of f .x/.
(b) g.a/ D 0.
(c) Df .a/ D 0.
9.3.4. Let K L be a field extension and let f .x/; g.x/ 2 KŒx. Show that the greatest common divisor of f .x/ and g.x/ in LŒx is the same as the greatest common divisor in KŒx. Hint: Review the algorithm for computing the g.c.d., using division with remainder.
9.3.5. Suppose f .x/ 2 KŒx, L is an extension field of K, and a is a multiple root of f .x/ in L. Show that if Df .x/ is not identically zero, then f .x/ and Df .x/ have a common factor of positive degree in LŒx and, therefore, by the previous exercise, also in KŒx.
9.3.6. Suppose that K is a field and f .x/ 2 KŒx is irreducible.
(a) Show that if f has a multiple root in some field extension, then Df .x/ D 0.
(b) Show that if Char.K/ D 0, then f .x/ has only simple roots in any field extension.
The preceding exercises establish the following theorem:
Theorem 9.3.1. If the characteristic of a field K is zero, then any irreducible polynomial in KŒx has only simple roots in any field extension.
9.3.7. If K is a field of characteristic p and a 2 K, then .x C a/p D xp C
p
ap. Hint: The binomial coefficient is divisible by p if 0