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Math Help - Help with two results on symmetric polynomials

  1. #1
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    Help with two results on symmetric polynomials

    Hi guys, I need your help again! I'm going through a proof in a textbook, and I can't prove the following two results

    1) Let \beta_1, \beta_2, \ldots, \beta_m be the roots of an equation dx^m+d_1x^{m-1} + \ldots + d_m = 0. Then, any symmetrical integral polynomial in d\beta_1, d\beta_2, \ldots, d\beta_m is an integral polynomial in d_1, d_2, \ldots, d_m and therefore an integer.

    2) Any symmetrical integral polynomial in d\alpha_1, \ldots, d\alpha_n is a symmetrical integral polynomial in d\alpha_1, \ldots, d\alpha_n, 0, 0, \ldots, 0.

    Thanks.
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  2. #2
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    Quote Originally Posted by Havelock View Post

    1) Let \beta_1, \beta_2, \ldots, \beta_m be the roots of an equation dx^m+d_1x^{m-1} + \ldots + d_m = 0. Then, any symmetrical integral polynomial in d\beta_1, d\beta_2, \ldots, d\beta_m is an integral polynomial in d_1, d_2, \ldots, d_m
    and therefore an integer.
    you should have mentioned that d, d_1, \cdots , d_m are integers. let \sigma_k(x_1, \cdots , x_m), \ 1 \leq k \leq m, be the k-th elementary symmetric polynomial. then we have:

    \sigma_k(d\beta_1, \cdots , d\beta_m)=d^k \sigma_k(\beta_1, \cdots , \beta_m)=d^k (-1)^k\frac{d_k}{d}=(-1)^kd^{k-1}d_k. now apply the fundamental theorem of symmetric polynomials to finish the proof.
    Last edited by NonCommAlg; December 25th 2008 at 12:09 PM.
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