On the Galois Theory of Differential Fields

Abstract

1. Summary. In a preceding paper [7] there was presented a theory, for a certaini kind of differenitial field extension called strongly normal. The of a strongly normal extension is enidowed with a structure very much like that of a variety, as studied by Weil [14]. In the present paper the study of such groups is renewed for the purpose of clarifyinig their conniection with varieties on the onie hand and with strongly normal extensiolis on the other. Consider a differential field S of characteristic 0 with algebraically closed field of constants W, and a strongly normal extension A of 5; suppose given a universal extension 5* of X, and denote the field of constants of 5* by AV. For reasons of convenience we use the term Galois of & over 7 for the of all (ipso facto strong) isomorphisms of & over 5, and not for its subgroup conisisting of all automorphisms of & over J. The field R*, being algebraically closed anid of finite transeendence degree over A, may be used as a uniiversal domain for geometry (Weil [13], ch. I, ? 1); it is the varieties of this geometry, defilned over E and slightly generalized to permit varieties which are reducible (i. e. which have more thani one comiponent), whieh we consider. By an algebraic group we miean either a or a variety as above. Chapter I is primarily a study of rational homomorphisms of groups into groups; these seenm to be the homomorphismls appropriate to the consideratioin of groups. Ratioinal homomorphisms of varities inlto varieties, without restrictioin to fields of eharacteristic 0, were considered by Weil [14] (who omitted the adjective rational). Specializing the coneept of rational homomorplhism we obtaini that of birational isomorphisin. In Chapter II it is showni that every is birationally isomorphic to a variety, and conversely that every irreducible variety is birationallv isomorphic to a group; this