Let $p$ be a prime, $n$ a positive integer and $q=p^n$ a $p$- power. Let $w,v$ is a root of the $q,p$-th cyclotomic polynomial $Q_q,Q_p$, respectively. Let $f$ be a polynomial with coefficient in $\mathbb Z$.
Prove or disprove: "$p^a\mid f(v) \implies p^a\mid f(w)$" in $\mathbb Z[w]$.
This question was an "iff" question (here), now it is "implies" question.
Moreover: $(p)=(1-v)^{p-1}$ in $\mathbb Z[v]$ and $(p)=(1-v)^{p^{n-1}(p-1)}$ in $\mathbb Z[w]$. That is, $p$ is totally ramified in both extension.