Picard-Vessiot Theory
In this chapter we give the basic algebraic results from the differential Galois theory of linear differential equations. Other presentations of some or all of this material can be found in the classics of Kaplansky [151] and Kolchin [162] (and Kolchin’s
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In this chapter we give the basic algebraic results from the differential Galois theory of linear differential equations. Other presentations of some or all of this material can be found in the classics of Kaplansky [151] and Kolchin [162] (and Kolchin’s original papers that have been collected in [25]) as well as the recent book of Magid [183] and the papers [231] and [173].
1.1 Differential Rings and Fields The study of polynomial equations leads naturally to the notions of rings and fields. For studying differential equations, the natural analogs are differential rings and differential fields, which we now define. All the rings considered in this chapter are assumed to be commutative, to have a unit element and to contain Q, the field of the rational numbers. Definition 1.1 A derivation on a ring R is a map ∂ : R → R having the properties that for all a, b ∈ R, ∂(a + b) = ∂(a) + ∂(b), and ∂(ab) = ∂(a)b + a∂(b). A ring R equipped with a derivation is called a differential ring and a field equipped with a derivation is called a differential field. We say a differential ring S ⊃ R is a differential extension of the differential ring R or a differential ring over R if the derivation of S restricts on R to the derivation of R. Very often we will denote the derivation of a differential ring by a → a . Furthermore, a derivation on a ring will also be called a differentiation. Examples 1.2 The following are differential rings. 1. Any ring R with trivial derivation, i.e., ∂ = 0. 2. Let R be a differential ring with derivation a → a . One defines the ring of differential polynomials in y1 , . . . , yn over R, denoted by R{{y1 , . . . , yn }},
M. van der Put. et al., Galois Theory of Linear Differential Equations © Springer-Verlag Berlin Heidelberg 2003
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1 Picard-Vessiot Theory ( j)
in the following way. For each i = 1, . . . , n, let yi , j ∈ N be an infinite set of distinct indeterminates. For convenience we will write yi for yi(0) , yi for yi(1) and yi for yi(2) . We define R{{y1 , . . . , yn }} to be the polynomial ring R[y1 , y1 , y1 , . . . , y2 , y2 , y2 , . . . , yn , yn , yn , . . . ]. We extend the derivation of R to ( j) ( j+1) a derivation on R{{y1 , . . . , yn }} by setting (yi ) = yi . Continuing with Example 1.2.2, let S be a differential ring over R and let ( j) ( j) u 1 , . . . , u n ∈ S. The prescription φ : yi → u i for all i, j, defines an R-linear differential homomorphism from R{{y1 , . . . , yn }} to S, that is φ is an R-linear homomorphism such that φ(v ) = (φ(v)) for all v ∈ R{{y1 , . . . , yn }}. This formalizes the notion of evaluating differential polynomials at values u i . We will write P(u 1 , . . . , u n ) for the image of P under φ. When n = 1 we shall usually denote the ring of differential polynomials as R{{y}}. For P ∈ R{{y}}, we say that P has order n if n is the smallest integer such that P belongs to the polynomial ring R[y, y , . . . , y(n) ]. Examples 1.3 The following are differential fields. Let C denote a field. 1. C(z), with derivation f → f = ddzf . 2.
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