  Let p be a prime and F(p) the maximal p-extension of a field F containing a
primitive p-th root of unity. We give a new characterization of Demuskin groups
among Galois groups Gal(F(p)/F) when p=2, and, assuming the Elementary Type
Conjecture, when p>2 as well. This characterization is in terms of the
structure, as Galois modules, of the Galois cohomology of index p subgroups of
Gal(F(p)/F).
