On abelian \(p\)-extensions of formal power series fields
Let \(k\) be a perfect field of characteristic a prime number \(p\) and \(k_{sep}^{[p]}\) be the maximal separable \(p\)-extension of \(k.\) Suppose that \(k\) is \(p\)-quasifinite, i.e., \(Gal(k_{sep}^{[p]}/k)\) is isomorphic to the ring of \(p\)-adic integers. If \(K=k((t_{n}))...((t_{1}))\) is a formal power series field in \(n\) variables. It is shown in this paper that for the Milnor \(K\)-group \(K_{n}^{M}K\) of \(K,\) the mapping from the set of finite abelian \(p\)-extensions \(L\) over \(K\) to the set of open subgroups of \(K_{n}^{M}K\) defined by \(L\rightarrow \overline{N_{L/K}K_{n}^{M}L}\) is an inclusion-reversing bijection, and \[ K_{n}^{M}K/\overline{N_{L/K}K_{n}^{M}L}\cong Gal(L/K). \] Moreover a necessary and sufficient condition for \(L/K\) to be unramified is also given, thereby generalizing some already proved results of the author (cf. [Tokyo J. Math. 6, 167-190 (1983; Zbl 0523.12015)]).
scientific article; zbMATH DE number 2145306
scientific article