On Generic differential $\operatorname{SO}_n$-extensions

Abstract

Let C be an algebraically closed field with trivial derivation and let F denote the differential rational field C , with Y ij , 1 < i < n - 1, 1 ≤ j ≤ n, i < j, differentially independent indeterminates over C. We show that there is a Picard-Vessiot extension e ⊃F for a matrix equation X' = XA.(Y ij ), with differential Galois group SO n , with the property that if F is any differential field with field of constants C, then there is a Picard-Vessiot extension E D F with differential Galois group H < SO n if and only if there are fij ∈ F with A(f ij ) well defined and the equation X' = XA(fij) giving rise to the extension E ⊃ F.