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# 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 Data provided are for informational purposes only. Although carefully collected, accuracy cannot be guaranteed. The impact factor represents a rough estimation of the journal's impact factor and does not reflect the actual current impact factor. Publisher conditions are provided by RoMEO. Differing provisions from the publisher's actual policy or licence agreement may be applicable.

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**ABSTRACT:**One of the fundamental questions in current field theory, related to Grothendieck's conjecture of birational anabelian geometry, is the investigation of the precise relationship between the Galois theory of fields and the structure of the fields themselves. In this paper we initiate the classification of additive properties of multiplicative subgroups of fields containing all squares, using pro-2-Galois groups of nilpotency class at most 2, and of exponent at most 4. This work extends some powerful methods and techniques from formally real fields to general fields of characteristic not 2. - [Show abstract] [Hide abstract]

**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.Journal of Algebra 10/2000; 235(2). DOI:10.1006/jabr.2000.8481 · 0.60 Impact Factor - [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.Journal of Pure and Applied Algebra 05/2001; 159(1-159):1-14. DOI:10.1016/S0022-4049(00)00117-1 · 0.47 Impact Factor