Generators and relations for the Brauer group of a field
The cup-product of the cohomology theory of groups induces a homomorphism \[ F^*\otimes X(F)=H^ 0(F,F^*_{sep})\otimes H^ 2(F,{\mathbb{Z}})\quad \to \quad H^ 2(F,F^*_{sep})=Br(F), \] where X(F) is the character group of the Galois group \(Gal(F_{sep}/F)\) and Br(F) is the Brauer group of the field F. The image of \(a\otimes \chi\) under this homomorphism turns out to be the class of a cyclic F-algebra (F(\(\chi\))/F,\(\sigma\),a). It is conjectured that this homomorphism is always surjective and its kernel is generated by all \(a\otimes \chi\), where \(a\in F^*\) is the norm in the extension F(\(\chi\))/F. More specifically, let \(S_ n(F)\) be the factor group of \(F^*\otimes_ nX(F)\) modulo the subgroup generated by all the ``symbols'' \(a\otimes \chi\), where a is the norm in F(\(\chi\))/F, and \({}_ nX(F)\) is the n-torsion of the group X(F). There is a homomorphism \(\alpha_ n: S_ n(F)\to_ nBr(F)\), and the conjecture states that \(\alpha_ n\) is an isomorphism for every n. This is shown to be the case in the following three cases: (1) \(n=p^ m\), where char F\(=p\neq 0\), (2) \(\mu_ n\subset F\), and (3) \(F(\mu_ n)/F\) is a quadratic extension of F, \(\mu_ n\) being the group of n-th roots of unity.
scientific article; zbMATH DE number 4212189