Pure Picard-Vessiot extensions with generic properties

Abstract. Given a connected linear algebraic group G over an algebraically closed eld C of characteristic 0, we construct a pure Picard-Vessiot extension for G, namely, a Picard-Vessiot extension E F, with dierential Galois group G, such that E and F are purely dierentially transcendental over C. The dierential eld E is the quotient eld of a G-stable proper dierential subring R with the property that if F is any dierential eld with eld of constants C and E F is a Picard-Vessiot extension with dierential Galois group a connected subgroup H of G, then there is a dierential homomorphism : R! E such that E is generated over F as a dierential eld by (R). 1