The Kummerian Property and Maximal Pro-$p$ Galois Groups

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 $\theta \text{\hspace{0.17em}⁣}:G\to 1+p{\mathbb{Z}}_p$, 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.