Simple algebras and cohomology groups of arbitrary fields

Abstract

Let F be a finite normal extension of a field C and let 9 be its Galois group of automorphisms. In the classical theory of simple algebras, it is shown that the Brauer group 23(F) of all simple algebras over C split by F is isomorphic (canonically) with the second group H2(F*, 9) of 9 with coefficients in the multiplicative group F* of the nonzero elements of F. If F is a purely inseparable extension of exponent 1 of C, then it was shown by N. Jacobson that a Galois theory for such extension is obtained by replacing the Galois group of automorphism of F by the restricted Liealgebra ? of all derivations of F over C. Recently, Hochschild has shown [7] that this Lie algebra can, in some way, replace the Galois group 9 in the classical result on the Brauer group !(F). More precisely: he has shown that !(F) is isomorphic with a certain subgroup of H*(F+, 3) where H*(F+, V) denotes the second restricted group of ? with coefficients in the additive group F+ of the elements of F. Naturally, the question arises now whether these two results are different results, which are connected by the name cohomology groups or whether they are two different aspects of one result which includes them. The main purpose of this paper is to answer this question in favor of the second possibility. A general Galois theory for arbitrary extensions F of C has been obtained by N. Jacobson [10], following some ideas of Kaloujnine, by considering representations of F over C. This idea has been followed, in an abstract form, considering F-bimodules and the results were extended to other cases by Hochschild [5], Nakayama [11; 12] and others. The general theory was obtained by considering certain relatively-cyclic sub-bimodules of the tensor product F0cF. On the other hand, the author has given a method for constructing all simple algebras (in [1]) by dealing with different imbeddings of a field F in a larger field K. An extension of this result to arbitrary rings R which is obtained with replacing the field K, in particular, by R = F0 GF, yields a relation between this construction of simple algebras and the general Galois theory. One aspect of this relation will be dealt with elsewhere. In the present paper we apply the extension of the results of [1 ] to obtain