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:
L =
W (y, z1,...,zd)
W (z1,...,zd)
where the Wronskian determinant is
(7) W (z1,...,zd) =
z1 . . . zd
z1 . . . zd
. . .
z1d−1) (
. . .
This argument is a straightforward imitation of the classical theory of algebraic field
extensions, where the Wronskian determinant plays the role of the Vandermonde.
L vanishes on all of V (z). Using the representation ρ, one can show that all the
coefficients of
L are left invariant under Gal∂(F/K). Hence,
L is a monic
linear differential operator with coefficients in K, whenever F/K is a weakly normal
extension. As d =
L ordK (z), we must have d =
L and
L = 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
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 =
is a unit of T (F/K),
= 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 +
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