Let $f \in \mathbb{Z}[X]$ be a monic polynomial of degree $d$. Let $E$ be the splitting field of $f$ over $\mathbb{Q}$ and let $R$ be the ring of integers in $E$. Suppose $p$ is a prime not dividing the discriminant $D_f$, let $\bar{f} \in \mathbb{F}_p[X]$ be its reduction modulo $p$ and let $P$ be a prime ideal of $R$ containing $p$. We then have that $R/P$ is a finite field of order dividing $p^n$.
Since $p \nmid D_f$, and $\overline{D_f}=D_{\bar{f}}$ is an integral polynomial in the coefficients of $f$, it follows that $f$ and $\bar{f}$ both have $d$ distinct roots in $R$ and $R/P$ respectively. Let $S_d$ denote the permutation group on $d$ elements. Then we have homomorphisms
$$\text{Gal}(E/\mathbb{Q}) \hookrightarrow S_d \leftarrow \text{Gal}((R/P)/\mathbb{F}_p)$$
Does anyone care to explain this last paragraph in more detail? Also, why is the arrow from $\text{Gal}((R/P)/\mathbb{F}_p)$ to $S_d$ not an inclusion?