Let $f(x)$ and $g(x)$ be irreducible polynomials over rationals. I know that to find the density of the primes $p$ such that (say) $f(x)$ has a root mod $p$ can be found using Chebotarev density theorem, looking at the density of elements with fixed points in the Galois group of $f(x)$. (Thinking of the Galois group embedded in $S_d$, where d is the degree of $f$) I was wondering what is the general way of doing this when we're given two polynomials $f$ and $g$, and if we try to find the density of primes such that both $f$ and $g$ have roots mod $p$. Should we somehow look at the compositum of the splitting fields of $f$ and $g$? That's my feeling, but I don't know how to do that. Thanks.
Let $P(h)$ be the proportion of primes $p$ such that the monic polynomial $h(t)$ has a root mod $p.$ You say that you can compute this when $h$ is irreducible. Your question is equivalent to asking if you can do it when $h$ is a product of two irreducibles:
You ask about finding $x,$ the proportion of primes where $f$ and $g$ both have roots. But $P(f)+P(g)=P(fg)+x.$