1-smooth pro-p groups and Bloch-Kato pro-p groups

Let p be a prime. A pro-p group G is said to be 1-smooth if it can be endowed with a homomorphism of pro-p groups of the form G → 1 + pZp satisfying a formal version of Hilbert 90. By Kummer theory, maximal pro-p Galois groups of fields containing a root of 1 of order p, together with the cyclotomic character, are 1-smooth. We prove that a finitely generated padic analytic pro-p group is 1-smooth if, and only if, it occurs as the maximal pro-p Galois group of a field containing a root of 1 of order p. This gives a positive answer to De Clercq–Florence’s “Smoothness Conjecture” — which states that the surjectivity of the norm residue homomorphism (i.e., the “surjective half” of the Bloch–Kato Conjecture) follows from 1-smoothness — for the class of finitely generated p-adic analytic pro-p groups.