12 GUILLAUME DUVAL To prove (ii) and (iii), denote by V (z) = VectC{σ(z)|σ Gal∂(F/K)}. We have a natural inclusion of V (z) in the solution space of Lz(y) = 0 in F : V (z) SolF (Lz(y) = 0) By the classical Wronskian argument (see below), we have: (6) dimC V (z) dimC(SolF (Lz(y) = 0)) ordK(z). Now (ii) follows from this inequality and (iii) will be a consequence of the equal- ity in (6). Set d = dimC V (z), and let {z = z1,...,zd} be conjugates of z under Gal∂(F/K), which form a C-basis of V (z). This basis gives rise to a linear repre- sentation: ρ : Gal∂(F/K) −→ GlC(V (z)). Consider the linear differential operator of order d: ˜(y) = W (y, z1,...,zd) W (z1,...,zd) where the Wronskian determinant is (7) W (z1,...,zd) = z1 . . . zd z 1 . . . z d . . . z(d−1) 1 . . . z(d−1) d . This argument is a straightforward imitation of the classical theory of algebraic field extensions, where the Wronskian determinant plays the role of the Vandermonde. ˜ vanishes on all of V (z). Using the representation ρ, one can show that all the coefficients of ˜(y) are left invariant under Gal∂(F/K). Hence, ˜(y) is a monic linear differential operator with coefficients in K, whenever F/K is a weakly normal extension. As d = ord(˜) ordK(z), we must have d = ord(˜) and ˜(y) = Lz(y) this gives the equality in (6) and completes the proof. (iv) is a result due to Harris and Sibuya, (see [29] or [27] ex 1.39, p. 30). (v) is Kolchin-Singer’s structure theorem for Picard-Vessiot field extensions. The letter “T” for the notation T (F/K) means that the affine scheme Spec(T (F/K)) is a G-torsor over K (see [27], Th 1.28, p. 22, Cor 1.30, p. 30 or [13], Th 5.12, p. 67). (vi) can be found in ([13] Prop 5.28, p. 73). Proposition 16. Let F/K be an ordinary differential field extension with the same constant field C. i. If w is a nonzero element of T (F/K) such that w w is a unit of T (F/K), then w is also a unit of T (F/K). ii. If F Gal∂(F/K) T (F/K), then T (F/K) is simple as a differential ring and as a Gal∂(F/K)-module. Proof. (i) Since u = w w is a unit of T (F/K), u u = α is algebraic over K by Harris- Sibuya’s theorem, (see Proposition 15(v)). Let’s compute the derivatives of w in terms of u and α: w = u · w w = u · w + u · w w = · u + u2)w
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