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College Algebra Exam Review 416

College Algebra Exam Review 416 - 426 9 FIELD EXTENSIONS...

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426 9. FIELD EXTENSIONS – SECOND LOOK The formal derivative in KOExŁ is defined by the usual rule from calcu- lus: We define D.x n / D nx n 1 and extend linearly. Thus, D. P k n x n / D P nk n x n 1 . The formal derivative satisfies the usual rules for differenti- ation: 9.3.1. Show that D.f .x/ C g.x// D Df .x/ C Dg.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.x p / . 9.3.3. Suppose f .x/ 2 KOExŁ , L is an extension field of K , and f .x/ factors as f .x/ D .x a/g.x/ in LOExŁ . 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 KOExŁ . Show that the greatest common divisor of f .x/ and g.x/ in LOExŁ is the same as the greatest common divisor in KOExŁ . Hint: Review the algorithm for
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