The Deuring-Šafarevič formula revisited (Q1080889)

Let K be an algebraic function field in one variable over an algebraically closed field k of characteristic \(p>0\). Let L/K be a finite p-extension. The relationship between the Hasse-Witt invariants of L and K is given by the Deuring-Shafarevich formula. Shafarevich established the formula when L/K is unramified. He then used it to show that the Galois group of the maximal unramified p-extension of K is a free pro-p- group on \(\lambda\) generators, \(\lambda\) being the Hasse-Witt invariant of K. In this paper we generalize Shafarevich's technique. Let S be a finite set of primes of K. For \(P\in S\) choose a finite p-extension \~K\({}_ P/K_ P\) with Galois group \(G_ P\). Let \(K_ T\) be the composite of all finite p-extensions L/K unramified outside of S and such that the local extension at any prime Q of L dividing a \(P\in S\) is isomorphic to a subfield of \~K\({}_ P/K_ P\). We use the Deuring-Shafarevich formula to show that the Galois group of \(K_ T/K\) is the p-profinite completion of the free product of the \(G_ P's\) and a free group on \(\lambda\) generators.