College Algebra Exam Review 432

College Algebra Exam Review 432 - Theorem 9.6.6. The set of...

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442 9. FIELD EXTENSIONS – SECOND LOOK Corollary 9.6.4. (a) Let f.x/ D x n C a n ± 1 x n ± 1 C ±±± C a 0 be a monic polynomial in KŒxŁ and let ˛ 1 ;:::;˛ n be the roots of f in a splitting field. Then a i D . ² 1/ n ± i ± n ± i 1 ;:::;˛ n / . (b) Let f.x/ D a n x n C a n ± 1 x n ± 1 C ±±± C a 0 2 KŒxŁ be of degree n , and let ˛ 1 ;:::;˛ n be the roots of f in a splitting field. Then a i =a n D . ² 1/ n ± i ± n ± i 1 ;:::;˛ n / . Proof. For part (a), f.x/ D .x ² ˛ 1 /.x ² ˛ 2 / ±±± .x ² ˛ n / D x n ² ± 1 1 ;:::;˛ n /x n ± 1 C ± 2 1 ;:::;˛ n /x n ± 2 ² ±±± C . ² 1/ n ± n 1 ;:::;˛ n /: For part (b), apply part (a) to .x ² ˛ 1 /.x ² ˛ 2 / ±±± .x ² ˛ n / D X i .a i =a n /x i : n Definition 9.6.5. Let K be a field and f u 1 ;:::;u n g a set of elements in an extension field. We say that f u 1 ;:::;u n g is algebraically independent over K if there is no polynomial f 2 KŒx 1 ;:::;x n Ł such that f.u 1 ;:::;u n / D 0 . The following is called the fundamental theorem of symmetric func- tions:
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Unformatted text preview: Theorem 9.6.6. The set of elementary symmetric functions f 1 ;:::; n g in Kx 1 ;:::;x n is algebraically independent over K , and gener-ates K S x 1 ;:::;x n as a ring. Consequently, K. 1 ;:::; n / D K S .x 1 ;:::;x n / . The algebraic independence of the i is the same as linear indepen-dence of the monic monomials in the i . First, we establish an indexing system for the monic monomials: A partition is a nite decreasing se-quence of nonnegative integers, D . 1 ;:::; k / . We can picture a parti-tion by means of an M-by-N matrix of zeroes and ones, where M k...
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