College Algebra Exam Review 431

College Algebra Exam Review 431 - 441 9.6. SYMMETRIC...

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Unformatted text preview: 441 9.6. SYMMETRIC FUNCTIONS I Proof. Exercise 9.6.2. Proposition 9.6.2. The field K.x1 ; : : : ; xn / of rational functions is Galois over the field K S .x1 ; : : : ; xn / of symmetric rational functions, and the Galois group AutK S .x1 ;:::;xn / .K.x1 ; : : : ; xn // is Sn . Proof. By Exercise 9.6.1, Sn acts on K.x1 ; : : : ; xn / by field automorphisms and K S .x1 ; : : : ; xn / is the fixed field. Therefore, by Proposition 9.5.5 the extension is Galois, with Galois group is Sn . I We define a distinguished family of symmetric polynomials, the elementary symmetric functions as follows: 0 .x1 ; : : : ; xn / D 1 1 .x1 ; : : : ; xn / D 2 .x1 ; : : : ; xn / D x1 C x2 C C xn X xi xj 1Äi <j Än ::: X D k .x1 ; : : : ; xn / x i1 xik 1Äi1 <i2 < <ik Än ::: D n .x1 ; : : : ; xn / We put j .x1 ; : : : xn / x1 x2 xn D 0 if j > n. Lemma 9.6.3. .x x1 /.x Dx D n n X 1x x2 /. n1 . 1/k C kx /.x 2x nk n2 xn / C . 1/n n ; k D0 where k is short for k .x1 ; : : : ; xn /. Proof. Exercise 9.6.4. I ...
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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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