College Algebra Exam Review 438

College Algebra - 448 9 FIELD EXTENSIONS – SECOND LOOK(a(b Œ.x1 x2.x1 x3.x1 x4.x2 x3.x2 m for D Œ4 3 3 1 in four variables x4.x3 x4/2

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Unformatted text preview: 448 9. FIELD EXTENSIONS – SECOND LOOK (a) (b) Œ.x1 x2 /.x1 x3 /.x1 x4 /.x2 x3 /.x2 m , for D Œ4; 3; 3; 1, in four variables x4 /.x3 x4 /2 9.6.15. Suppose that f is an antisymmetric polynomial in n variables; that is, for each 2 Sn , .f / D . /f , where denotes the parity homomorphism. Show Q f has a factorization f D ı.x1 ; : : : ; xn /g , that where ı.x1 ; : : : ; xn / D i <j .xi xj /, and g is symmetric. 9.7. The General Equation of Degree n Consider the quadratic formula or Cardano’s formulas for solutions of a cubic equation, which calculate the roots of a polynomials in terms of the coefficients; in these formulas, the coefficients may be regarded as variables and the roots as functions of these variables. This observation suggests the notion of the general polynomial of degree n, which is defined as follows: Let t1 ; : : : ; tn be variables. The general (monic) polynomial of degree n over K is Pn .x/ D x n t1 x n 1 C C . 1/n 1 tn 2 K.t1 ; : : : ; tn /.x/: Let u1 ; : : : ; un denote the roots of this polynomial in a splitting field E . Then .x u1 / .x un / D Pn .x/; and tj D j .u1 ; : : : ; un / for 1 Ä j Ä n, by Corollary 9.6.4. We shall now show that the Galois group of the general polynomial of degree n is the symmetric group Sn . Theorem 9.7.1. Let E be a splitting field of the general polynomial Pn .x/ 2 K.t1 ; : : : ; tn /Œx. The Galois group AutK.t1 ;:::;tn / .E/ is the symmetric group Sn . Proof. Introduce a new set of variables v1 ; : : : ; vn and let fj D j .v1 ; : : : ; vn / for 1 Ä j Ä n; where the j are the elementary symmetric functions. Consider the polynomial X Q Pn .x/ D .x v1 / .x vn / D x n C . 1/j fj x n j : j The coefficients lie in K.f1 ; : : : ; fn /, which is equal to K S .v1 ; : : : ; vn / by Theorem 9.6.6. According to Proposition 9.6.2, K.v1 ; : : : ; vn / is Galois over K.f1 ; : : : ; fn / with Galois group Sn . Furthermore, Q K.v1 ; : : : ; vn / is the splitting field over K.f1 ; : : : ; fn / of Pn .x/. Let u1 ; : : : ; un be the roots of Pn .x/ in E . Then tj D j .u1 ; : : : ; un / for 1 Ä j Ä n, so E D K.t1 ; : : : ; tn /.u1 ; : : : ; un / D K.u1 ; : : : ; un /. ...
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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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