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 θ:G→1+pZ _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.