Suppose $f(x) \in \mathbb{Z}[x]$ is an irreducible monic polynomial over $\mathbb{Q}(\alpha)$ of degree $n$, where $\alpha$ is a root of a monic polynomial $g(x) \in \mathbb{Z}[x]$. Assume that the Galois group of $f$ is nice enough (for example, it has an $n$-cycle). Do there exist infinitely many primes $p$ such that $f$ stays irreducible over $\mathbb{F}_p(\alpha)$? By $\mathbb{F}_p(\alpha)$ I mean adjoin to $\mathbb{F}_p$ any root of $g$ ($g$ might not be irreducible anymore). This might follow from Chebotarev's theorem, and I would be happy if someone could confirm / explain it to me.
Furthermore, do there exist infinitely many primes $p$ such that $f$ stays irreducible over $\mathbb{F}_p$, but $g$ splits completely in $\mathbb{F}_p$?