College Algebra Exam Review 415

College Algebra Exam Review 415 - 425 9.3. THE DERIVATIVE...

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Unformatted text preview: 425 9.3. THE DERIVATIVE AND MULTIPLE ROOTS Q Corollary 9.2.5. Let p.x/ 2 KŒx, and suppose L and L are two splitting Q fields for p.x/. Then there is an isomorphism W L ! L such that .k/ D k for all k 2 K . Q Proof. Take K D K and D id in the proposition. I 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 x 6 3 is irreducible over Q. It factors over Q.31=6 / as .x If ! D e is .x 31=6 /.x C 31=6 /.x 2 i=3 , then the irreducible factorization of f .x/ over Q.31=6 ; !/ 31=6 /.x C 31=6 /.x 9.2.3. (a) (b) (c) 31=6 x C 31=3 /.x 2 C 31=6 x C 31=3 /: ! 31=6 /.x ! 2 31=6 /.x ! 4 31=6 /.x ! 5 31=6 /: 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. Show that if K is a finite field, then K admits field extensions of arbitrarily large finite degree. 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/m g.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. ...
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