On Splitting Fields of Simple Algebras

J. H. M. Wedderburn,l in the course of his fundamental investigations on associative algebras proved that if A is a central2 simple algebra over the field F, there exist extension fields K of F such that the extension AR C A X K of A to the ground field K is isomorphic to a complete matric algebra of a certain degree n over K. The fields K with this property have been called the splitting fields of A. Since A is contained in AK, it follows that A possesses an absolutely irreducible representation by matrices with coefficients in K, and the splitting fields of A can be characterized by this fact. This already shows the close relation between Wedderburn's theory and I. Schur's investigations on the representations of (semi-) groups by linear transformations. This connection was studied later more closely by Emmy Noether and the author.3 It was shown that the splitting fields of finite degree of A can be characterized as the maximal (commutative) subfields of the algebras B of the class of A. Here, two simple algebras A and A1 over F belong to the same class, if in their Wedderburn decompositions A = A X M, Al = Al X M, as direct products of division algebras A, A1 and complete matric algebras M, M, the factors A and A1 are isomorphic. If the Wedderburn factor A of A has the rank m2 over F, m is called the index of the algebra class of A. Then m divides the degree of any splitting field of A, and there exist splitting fields of degree m. When we start now from an arbitrary extension field K of degree n over F, then K may or may not4 be a splitting field of central simple algebras A of a given index m dividing n. In this manner, the theory of algebras provides a classification of extension fields K of a given field. The problem arises to discuss the significance of this classification from the viewpoint of the theory of fields and their Galois groups. That this is possible can be expected because, roughly speaking, the study of simple algebras is equivalent to an investigation of field theoretical questions, as was shown by the author in an earlier paper.' Even when K is a normal field over F, the answer to our question cannot be given in