Let $p$ be a rational prime and $K$ a number field. Dedekind's discriminant theorem tells us that $p$ ramifies in $K$ $\iff$ $p$ divides the discriminant of $K$. Hence if $p$ does not divide discriminant of $K$, $(p)$ either splits, i.e.,

(i) $(p)=P_1 \cdots P_g$ for $P_i \neq P_j$ and $g \geq 2$ or

(ii) $(p)$ remains prime.

Now, my question is: what are some criteria which can tell if $p$ will split or remain prime?