  We prove that two arithmetically significant extensions of a field F coincide
if and only if the Witt ring WF is a group ring Z/n[G]. Furthermore, working
modulo squares with Galois groups which are 2-groups, we establish a theorem
analogous to Hilbert's Theorem 90 and show that an identity linking the
cohomological dimension of the Galois group of the quadratic closure of F, the
length of a filtration on a certain module over a Galois group, and the
dimension over Z/2 of the square class group of the field holds for a number of
interesting families of fields. Finally we discuss the cohomology of a
particular Galois group in a topological context.
