Generic Splitting Fields of Central Simple Algebras

if and only if 5F splits W. The extension of this result is the main object of the present paper. In contrast with the method of Witt, which uses the theory of the arithmetics of these function fields, the present method aims to obtain a generic condition forafield to split a given c.s.a. 2I. It is shown that to every c.s.a. ? of order n over e there corresponds an algebraic function field of n - 1 variables D(S[) over e which splits WI. D(W1) is the most general splitting field of ?1 in the sense that the residue fields of D(SL) are the splitting fields of SI. The extension of the genus zero is the fact that D(W[) is almost rational. In fact, the splitting fields & of ?I are characterized as those constant field extension E of D(S) which turn D(W1) into a field of all rational function in n- 1 variables over e. The correspondence ? + D(SI) is not unique, it seems that D(S[) depends on W as well as on its powers. As a result of constructing the fields D(W1) we show that the splitting fields of a c.s.a. ?I split also a c.s.a. !B if and only if Z is similar in the sense of Brauer to some power of W. The main tools for studying these fields are obtained in the first part of the present paper. Here, a generalization of the theory of representation of c.s.a. as developed by R. Brauer in [3 and 4 is given. Brauer has studied relation of the matrix representation of a given c.s.a. ? in an algebraic splitting field 5f and the conjugates of this representation in the matrix ring over the normal field containing 5. The present generalization lies in the fact that the study is not carried only for algebraic fields but also for transcendal fields and their isomorphisms. This enables one to obtain generic condition for arbitrary splitting fields. The present method is carried in linear transformation of vectors spaces instead of matrices with the aid of generalizing the notion of semi linear transformation. Additional applications of the method developed in the first part will be given in a subsequent paper. The subject of correspondence between c.s.as. and transcendental field is related to a correspondence between c.s.as. and certain algebraic varieties dealt