let p be a rational prime,K a number field. Dedekind's discriminant theorem tells us: p ramifies in K <==> p divides discriminant of K. hence if p does not divide discriminant of K, (p) will either splits,i.e.(p)=P_1P_2 (P1=/= P2) or  (p) remains prime. now,my question is  : there are some criterias which can tell if p will split or remain prime.

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?