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 $\displaystyle 1\leq i<j\leq n$?