Factoring a prime over a splitting field and factoring a polynomial over of a finite field Let $f$ be a separable monic polynomials with integer coefficient. Let $K$ be the splitting field of $f$ over $\mathbb{Q}$, and let $p$ be a prime. What is the relationship between the statements 
$f$ splits over $\mathbb{F}_p$ and 
$p$ is not irreducible in $K$ (there is probably some fancy term for this, like $p$ is inert in $K$?)
There are equivalent for the Gaussian Integers, so probably also for any quadratic extension with trivial class group.
I suspect they are equivalent, and this is probably extremely well known, but I have asked a couple people with no luck.

Let’s look at a simple example, $f(X)=X^3-2$ and its splitting field $K=\Bbb Q(\lambda,\omega)$, over $\Bbb Q$, where $\lambda=\sqrt[3]2$ and $\omega$ is a primitive cube root of unity. Take $p=5$, and look at $f\in\Bbb F_5[X]$, where it factors as $(X+2)(X^2+3X+4)$, so $f$ doesn’t split there.
Calling $\mathscr O$ the ring of algebraic integers of $K$, we can see that $5\mathscr O$ factors as $\mathfrak p_1\mathfrak p_2\mathfrak p_3$, where in each case $\mathscr O/\mathfrak p_i\cong\Bbb F_{25}$.
