Help with two results on symmetric polynomials

**Havelock** · December 25th 2008, 07:53 AM

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.

**NonCommAlg** · December 25th 2008, 11:22 AM

Originally Posted by Havelock

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.

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