Galois extensions with bounded ramification in characteristic \(p\). On a question of S. Abhyankar (Q1914853)

The purpose of this paper is to give a partial answer to a question raised by S. Abhyankar in 1995 concerning the realizability of certain groups as Galois groups of extensions of congruence function fields with restricted ramification. Thus, the authors prove in six steps, using as main ingredients the Chebotarev density theorem, the Carlitz-Hayes theory of cyclotomic function fields and the Riemann-Roch theorem, the following result: Let \(G\) be a finite group with a normal abelian \(p\)-Sylow subgroup \(p(G)\) such that \(G/p(G)\) is nilpotent. Let \(K/k\) be a congruence function field of characteristic \(p\), \(q\) the cardinality of \(k\), and \(s\) the minimum number of generators of \(G/p(G)\). Then, there exists a Galois extension \(L/k\) of \(K/k\) such that \(\text{Gal} (L/K)\) is isomorphic to \(G\) and precisely \(s\) prime divisors of \(K\) are ramified.