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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 KOExŁ and let ˛ 1 ; : : : ; ˛ n be the roots of f in a splitting field. Then a i D . 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 KOExŁ be of degree n , and let ˛ 1 ; : : : ; ˛ n be the roots of f in a splitting field. Then a i =a n D . 1/ n i n i 1 ; : : : ; ˛ n / . Proof. For part (a), f .x/ D .x ˛ 1 /.x ˛ 2 / .x ˛ n / D x n 1 1 ; : : : ; ˛ n /x n 1 C 2 1 ; : : : ; ˛ n /x n 2 C . 1/ n n 1 ; : : : ; ˛ n /: For part (b), apply part (a) to .x ˛ 1 /.x ˛ 2 / .x ˛ 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 KOEx 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 KŒx 1 ;:::;x n Ł is algebraically independent over K , and gener-ates K S Œx 1 ;:::;x n Ł 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 ﬁnite decreasing se-quence of nonnegative integers, ² D .² 1 ;:::;² k / . We can picture a parti-tion by means of an M-by-N matrix ³ of zeroes and ones, where M ³ k...
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