Monic polynomials with integer coefficients

Question

We have $\Pi_{j=1}^n (z-z_j)$ a polynomial with integer coefficients. Is also $\Pi_{j=1}^n (z-z_j^k)$ for k=1,2,3,... a polynomial with integer coefficients?

Answer 1

Yes. The coefficients of the latter are symmetric polynomials in the $z_j$, hence are (integer!) polynomials in the elementary symmetric polynomials in the $z_j$, that is in the coefficients of the former polynomial.