So let $f(x)\in\mathbf{Z}[x]$ be a monic polynomial of degree $d$ and let $K$ be the splitting field of $f$. Let us define
the "heigt of $f$" $:=||f||$ to be the maximum of the abolute values of
the coefficients of $f$. (Instead of the height it might be better to work with the abolute value of the discriminant of $K$).
Let us denote the Galois group of $f$ over $\mathbf{Q}$ by
$G$. For each prime number for which the roots of $f$ modulo $p$ are distinct we denote by
$G_p$ the Galois group of $f\pmod{p}$. A cute result that may be found for example in Van der Waerden first algebra book says that there exists an (non-canonical but well defined up to conjugation in $G$) injection of $G_p$ in $G$. By elementary group theory, if we take the group generated by a set of representatives of the conjugacy classes of $G$ then it generates $G$. Thus by Chebotarev density theorem, we know that there exists a finite set of prime numbers
$S$ of $\mathbf{Q}$ such that
$$
G_S:=\langle G_p| p\in S\rangle=G.
$$
In particular, we may always choose a set $S$ with $G_S=G$ and
$|S|\leq r$ where $r$ is the number of conjugacy classes of $G$.
Q: So let $S_x$ be the set of all prime numbers less than $x$. Is it possible to find explicitly a lower bound for $x$ in terms $||f||$ (or $|disc(K)|$) and $d$ such that
$$
G_{S_x}=G?
$$
added: So basically, I'm just asking for an effective version of the Chebotarev density theorem for the splitting field of $f$, this is probably well known to the expert.
So probably one should consider $|disc(K)|$ rather than $||f||$ which can be arbitrary large for a fixed $K$ (even though $f$ may have bad reduction for many primes $p$ which do not divide $|disc(K)|$)
Best Answer
You are looking for the very useful paper
Effective versions of the Chebotarev density theorem, J. C. Lagarias and A. M. Odlyzko, pp. 409-464 in Algebraic Number Fields, A. Frohlich (ed.), Academic Press, 1977.
The bounds there are quite large, as I recall, especially if you don't want to assume GRH. There is very nice recent work of Jouve, Kowalski, and Zywina that give a lot of insight into how to do this in practice and how many prime numbers you shoudl expect to have to use: see Kowalski's blog post about this and work cited there.