Article

# On the Cohomology of Galois Groups Determined by Witt Rings

Department of Mathematics, University of Western Ontario, London, Ontario, Canada, N6A 5B7f2E-mail: minac@uwo.caf2

Advances in Mathematics (Impact Factor: 1.29). 08/1998; 148(1):105-160. DOI: 10.1006/aima.1999.1847 Get notified about updates to this publication Follow publication |

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**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. - [Show abstract] [Hide abstract]

**ABSTRACT:**In this paper we provide calculations for the cohomology of certain p-groups, using topological methods. More precisely, we look at p-groups G defined as central extensions 1→V→G→W→1 of elementary abelian groups such that and the defining k-invariants span the entire image of the Bockstein. We show that if p>dimV−dimW+1, then the cohomology of G can be explicitly computed as an algebra of the form where is a polynomial ring on two-dimensional generators and A is the cohomology of a compact manifold which in turn can be computed as the homology of a Koszul complex. As an application we provide a complete determination of the cohomology of the universal central extension provided , where n=dimW.