  We observe that some basic but fundamental constructions in Galois theory can
be used to obtain some interesting restrictions on the structure of Galois
groups of maximal $p$-extensions of fields containing a primitive $p$th root of
unity. This is an extension of some significant ideas of Demushkin, Labute and
Serre from local fields to all fields containing a primitive $p$th root of
unity. Our techniques use certain natural simple Galois extensions together
with some considerations in Galois cohomology and Massey products.
