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 $|disc(K)|$)



Effective Chebotarev density results for arbitrary number fields 
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](http://blogs.ethz.ch/kowalski/2008/07/24/the-chebotarev-invariant-or-wait-and-see/) and work cited there. 


