Galois-theoretic features for 1-smooth pro-<i>p</i> groups

Abstract Let p be a prime. A pro- group G is said to 1-smooth if it can endowed with continuous representation $\theta \colon G\to \mathrm {GL}_1(\mathbb {Z}_p)$ such that every open subgroup H of , together the restriction \vert _H$ satisfies formal version Hilbert 90. We prove contains unique maximal closed abelian normal subgroup, in analogy result by Engler and Koenigsmann on Galois groups fields, solvable, then locally uniformly powerful, Ware fields. Finally, we ask whether satisfy “Tits’ alternative.”