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The minimal ramification problem may be considered as a quantitative versionof the inverse Galois problem. For a nontrivial finite group $G$, let $m(G)$ bethe minimal integer $m$ for which there exists a Galois extension$N/\mathbb{Q}$ that is ramified at exactly $m$ primes (including the infiniteone). So, the problem is to compute or to bound $m(G)$.  In this paper, we bound the ramification of extensions $N/\mathbb{Q}$obtained as a specialization of a branched covering $\phi\colon C\to\mathbb{P}^1_{\mathbb{Q}}$. This leads to novel upper bounds on $m(G)$, forfinite groups $G$ that are realizable as the Galois group of a branchedcovering. Some instances of our general results are: $$ 1\leq m(S_m)\leq 4\quad \mbox{and} \quad n\leq m(S_m^n) \leq n+4, $$ for all $n,m>0$. Here $S_m$denotes the symmetric group on $m$ letters, and $S_m^n$ is the direct productof $n$ copies of $S_m$. We also get the correct asymptotic of $m(G^n)$, as $n\to \infty$ for a certain class of groups $G$.  Our methods are based on sieve theory results, in particular on theGreen-Tao-Ziegler theorem on prime values of linear forms in two variables, onthe theory of specialization in arithmetic geometry, and on finite grouptheory.

Sieves and the Minimal Ramification Problem