  For a prime number $p$, we give a new restriction on pro-$p$ groups $G$ which
are realizable as the maximal pro-$p$ Galois group $G_F(p)$ for a field $F$
containing a root of unity of order $p$. This restriction arises from Kummer
Theory and the structure of the maximal $p$-radical extension of $F$. We study
it in the abstract context of pro-$p$ groups $G$ with a continuous homomorphism
$\theta\colon G\to1+p\mathbb{Z}_p$, and characterize it cohomologically, and in
terms of 1-cocycles on $G$. This is used to produce new examples of pro-$p$
groups which do not occur as maximal pro-$p$ Galois groups of fields as above.
