Showing papers on "Generic polynomial published in 1964"

On the inverse problem of Galois theory of differential fields

Abstract: 1. All fields considered here are of characteristic 0. Let F be a field, let C be an algebraically closed subfield of F. Let G be a connected algebraic group defined over C. F(G) denotes the field of all rational functions on G defined over F. If gCG then F(g) denotes the field generated by g over F. We shall say that a derivation of F(G) commutes with G*(C) if it commutes with g*, for every gEG(C), where g* denotes the automorphism of F(G) induced by the left translation by g, i.e., (g*f)(x) =f(gx), for any xCG. F denotes the Lie algebra of all derivations of F(G) that are zero on F and which commute with G*(F). If G1 is a normal subgroup of G defined over F then F(G/G1) is canonically isomorphic to a subfield of F(G); we shall identify F(G/G1) and this subfield. If R is an integral domain then (R) denotes the field of fractions of R. Every derivation d of R can be uniquely extended to a derivation of R (the extended derivation will be also denoted by d). If F1, F2 are two fields containing F as a subfield and if d1, d2 are derivations of F1, F2, respectively, such that d1i F= d2 I F and d1(F) C F then d1i d2 denotes the derivation of F1,0F F2 determined by (d1 d2)(a 0 b) -d1(a)Gb+a0d2(b), for every aGF1 and bCF2. do denotes the zero derivation of a field (it will be always clear what field we have in mind). The underlying field of an ordinary differential field 53 will be denoted by F.