1. ## Elementary symmetric polynomials

Let elementary symmetric polynomials (Vieta's formula)
$\displaystyle \Pi_k(x_1,x_2,\ldots x_n)$ be integers for every $\displaystyle k=\overline{1,n}$. Is the polynomial
$\displaystyle \Pi_2(x_1x_2,x_1x_3\ldots ,x_ix_j,\ldots x_{n-1}x_n)$ also integer, where $\displaystyle 1\leq i<j\leq n$?

2. Originally Posted by rodeo
Let elementary symmetric polynomials (Vieta's formula)
$\displaystyle \Pi_k(x_1,x_2,\ldots x_n)$ be integers for every $\displaystyle k=\overline{1,n}$. Is the polynomial
$\displaystyle \Pi_2(x_1x_2,x_1x_3\ldots ,x_ix_j,\ldots x_{n-1}x_n)$ also integer, where $\displaystyle 1\leq i<j\leq n$?
In terms of the elementary symmetric polynomials $\displaystyle E_k = \Pi_k(x_1,x_2,\ldots x_n)$, it looks as though $\displaystyle \Pi_2(x_1x_2,x_1x_3,\ldots ,x_ix_j,\ldots x_{n-1}x_n) = E_1E_3 - E_4$. So it will be an integer.

3. I think that I prove this by an easy induction.
What about $\displaystyle \Pi_k(x_1x_2,x_1x_3,\ldots,x_ix_j,\ldots,x_{n-1}x_n )$ for any $\displaystyle 2 \leq k \leq n(n-1)/2$?

Thank you!