College Algebra Exam Review 439

College Algebra Exam Review 439 - ± n.x i ± x j We know...

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9.7. THE GENERAL EQUATION OF DEGREE N 449 Since f t i g are variables, and the f f i g are algebraically independent over K , according to Theorem 9.6.6 , there is a ring isomorphism KŒt 1 ;:::;t n Ł ! KŒf 1 ;:::;f n Ł fixing K and taking t i to f i . This ring isomorphism extends to an isomorphism of fields of fractions K.t 1 ;:::;t n / Š K.f 1 ;:::;f n /; and to the polynomial rings K.t 1 ;:::;t n /ŒxŁ Š K.f 1 ;:::;f n /ŒxŁ I the isomorphism of polynomial rings carries P n .x/ to Q P n .x/ . Therefore, by Proposition 9.2.4 , there is an isomorphism of splitting fields E Š K.v 1 ;:::;v n / extending the isomorphism K.t 1 ;:::;t n / Š K.f 1 ;:::;f n /: It follows that the Galois groups are isomorphic: Aut K.t 1 ;:::;t n / .E/ Š Aut K.f 1 ;:::;f n / .K.v 1 ;:::;v n // Š S n : n We shall see in Section 10.6 that this result implies that there can be no analogue of the quadratic and cubic formulas for equations of degree 5 or more. (We shall work out formulas for quartic equations in Section 9.8 .) The discriminant. We now consider some symmetric polynomials that arise in the study of polynomials and Galois groups. Write ı D ı.x 1 ;:::;x n / D Y 1 ± i<j
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Unformatted text preview: ± n .x i ± x j /: We know that ı distinguishes even and odd permutations. Every permuta-tion ± satisfies ±.ı/ D ˙ ı and ± is even if, and only if, ±.ı/ D ı . The symmetric polynomial ı 2 is called the discriminant polynomial . Now let f D P i a i x i 2 KŒxŁ be polynomial of degree n . Let ˛ 1 ;:::;˛ n be the roots of f.x/ in a splitting field E . The element ı 2 .f / D a 2n ² 2 n ı 2 .˛ 1 ;:::;˛ n / is called the discriminant of f . Now suppose in addition that f is irreducible and separable. Since ı 2 .f / is invariant under the Galois group of f , it follows that ı 2 .f / is an element of the ground field K . Since the roots of f are distinct, the element ı.f / D a n ² 1 n ı.˛ 1 ;:::;˛ n / is nonzero, and an element ± 2 Aut K .E/ induces an even permutation of the roots of f if, and only if, ±.ı.f // D ı.f / . Thus we have the following result:...
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