  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 $G\to1+p\mathbb{Z}_p$ 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 $p$-adic 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 Clerq-Florence's "Smoothness Conjecture" - which states
that the Rost-Voevodsky Theorem (a.k.a. Bloch-Kato Conjecture) follows from
1-smoothness - for the class of finitely generated $p$-adic analytic pro-$p$
groups.
