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College Algebra Exam Review 438

College Algebra Exam Review 438 - 448 9 FIELD EXTENSIONS...

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448 9. FIELD EXTENSIONS – SECOND LOOK (a) OE.x 1 x 2 /.x 1 x 3 /.x 1 x 4 /.x 2 x 3 /.x 2 x 4 /.x 3 x 4 2 (b) m , for D OE4; 3; 3; 1Ł , in four variables 9.6.15. Suppose that f is an antisymmetric polynomial in n variables; that is, for each 2 S n , .f / D . /f , where denotes the parity homomorphism. Show that f has a factorization f D ı.x 1 ; : : : ; x n /g , where ı.x 1 ; : : : ; x n / D Q i<j .x i x j / , 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 vari- ables and the roots as functions of these variables. This observation sug- gests the notion of the general polynomial of degree n , which is defined as follows: Let t 1 ; : : : ; t n be variables. The general (monic) polynomial of degree n over K is P n .x/ D x n t 1 x n 1 C C . 1/ n 1 t n 2 K.t 1 ; : : : ; t n /.x/: Let u 1 ; : : : ; u n denote the roots of this polynomial in a splitting field E . Then .x u 1 / .x u n / D P n .x/; and t j D j .u 1 ; : : : ; u n / for 1 j n , by Corollary 9.6.4 . We shall now show that the Galois group of the general polynomial of degree
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