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)

L =

W (y, z1,...,zd)

W (z1,...,zd)

where the Wronskian determinant is

(7) W (z1,...,zd) =

z1 . . . zd

z1 . . . zd

. . .

z1d−1) (

. . .

zdd−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

˜(y)

L are left invariant under Gal∂(F/K). Hence,

˜(y)

L is a monic

linear differential operator with coeﬃcients in K, whenever F/K is a weakly normal

extension. As d =

ord(˜)

L ordK (z), we must have d =

ord(˜)

L and

˜(y)

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

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 = (α · u +

u2)w