On page 35 in Niven's book on irrationals , during the proof of Lemma 3.6: That if $\omega$ is a primitive $n$th root of unity, and $f(x)$ is it's minimal polynomial, then for any prime $p$ such that $(p,n) = 1$ and any $\rho$ such that $f(\rho) = 0$ we have that $f(\rho^p) = 0,$ the author makes the following two statements:
1."By the fundamental theorem on symmetric polynomials, the $p$th powers of the roots of $f(x) = 0$ are roots of a polynomial equation which is also monic with integral coefficients, and so $\rho^p$ is an algebraic integer."
How exactly are we applying the fundamental theorem of symmetric polynomials here?
2."We now interpret these equations in $J_p[x]$." ($J_P[x]$ are the polynomials with coefficients in the field of integers modulo $p$).
Why exactly are we allowed to do that? Wouldn't that imply that the polynomials used in the aboive equations already have coefficients in $J_p[x]$?