Field Theory and the Cohomology of Some Galois Groups

Alejandro Adem
Mathematics Department
University of Wisconsin
Madison, Wisconsin, 53706 

Wenfeng Gao
4426 S. 112th E. Ave. 
Tulsa, OK

Dikran B. Karagueuzian
Mathematics Department
University of Wisconsin
Madison, Wisconsin, 53706 

Jan Minac
Mathematics Department 
University of Western Ontario
London, Ontario, Canada N6A 5B7


Abstract

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.