I was wondering what general strategies are available to figure out if a prime splits? I know for quadratic extensions there aren't too many possibilities for how a prime can split, so we essentially only need to check that $X^2-d$ has a root modulo $p$ and that $p$ does not divide the discriminant. This can be one using quadratic reciprocity.
For e.g. an extension $\mathbb{Q}(\alpha)/\mathbb{Q}$, where $\alpha^3-\alpha-1=0$ there are a few more options. In this particular example computing the discriminant is easy and in general there are algorithms for it. The question is that for a prime $\mathfrak{P}\mid p$ how do we know that $f(\mathfrak{P}/p)=1$? Is there any "simple" algorithm that works for the polynomials $X^3+aX+b$ and $X^5+aX+b$?
EDIT: Per instructions below I'm editing the question. I was more interested in actually classifying and finding the density of the set of primes with a particular factorization. Given a particular prime, its factorization is much easier to find. In the quadratic case this is done by using quadratic reciprocity as I described above. What I don't know is how to work with cubics. I've been told that this can be done for cubics $X^3+aX+b$ and for simplicity I'm interested in the case $X^3-X-1$. Apparently another set of "easier" equations is $X^5+aX+b$.