College Algebra Exam Review 416

College Algebra - 426 9 FIELD EXTENSIONS – SECOND LOOK The formal derivative in KŒx is defined by the usual rule P calcufrom lus We define

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Unformatted text preview: 426 9. FIELD EXTENSIONS – SECOND LOOK The formal derivative in KŒx is defined by the usual rule P calcufrom lus: We define D.x n / D nx n 1 and extend linearly. Thus, D. kn x n / D P nkn x n 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) (b) 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. 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.x p /. 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) (b) (c) a is a multiple root of f .x/. g.a/ D 0. 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) (b) Show that if f has a multiple root in some field extension, then Df .x/ D 0. 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. ...
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This note was uploaded on 12/15/2011 for the course MAC 1105 taught by Professor Everage during the Fall '08 term at FSU.

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