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For a prime number $p$, we give a new restriction on pro-$p$ groups $G$ whichare 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 KummerTheory and the structure of the maximal $p$-radical extension of $F$. We studyit 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 interms 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.

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