Lectures on Differential Galois TheoryDifferential Galois theory studies solutions of differential equations over a differential base field. In much the same way that ordinary Galois theory is the theory of field extensions generated by solutions of (one variable) polynomial equations, differential Galois theory looks at the nature of the differential field extension generated by the solution of differential equations. An additional feature is that the corresponding differential Galois groups (of automorphisms of the extension fixing the base and commuting with the derivation) are algebraic groups. This book deals with the differential Galois theory of linear homogeneous differential equations, whose differential Galois groups are algebraic matrix groups. In addition to providing a convenient path to Galois theory, this approach also leads to the constructive solution of the inverse problem of differential Galois theory for various classes of algebraic groups. Providing a self-contained development and many explicit examples, this book provides a unique approach to differential Galois theory and is suitable as a textbook at the advanced graduate level. |
Contents
CHAPTER 1 Differential Ideals | 1 |
CHAPTER 2 The Wronskian | 15 |
CHAPTER 3 PicardVessiot Extensions | 23 |
CHAPTER 4 Automorphisms of PicardVessiot Extensions | 43 |
CHAPTER 5 The Structure of PicardVessiot Extensions | 61 |
CHAPTER 6 The Galois Correspondence and its Consequences | 75 |
CHAPTER 7 The Inverse Galois Problem | 89 |
| 103 | |
Back Cover | 104 |
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Common terms and phrases
algebraic group structure algebraic subgroup algebraically closed field basis coefficients coordinate ring Corollary denote derivation differential F-algebra differential field differential field extension differential Galois theory differential integral domain differential ring differential subfield elements Example extension E D F extension of F field extension field F field of constants field with algebraically FIGF follows fraction field full universal solution G E/F G s G(E/F G-stable ideals Galois extension GL(V group G group of differential hence homogeneous linear differential implies integral domain intermediate differential field isomorphism kernel Lemma let E D F Let F Let G linear differential operator linear homogeneous differential linearly independent maximal differential ideal monic homogeneous linear non-zero normal subgroup notation numbers Picard–Vessiot extension polynomial ring PROOF proper differential ideals Proposition 3.9 quotient rational function set of solutions subalgebra Suppose surjective Theorem 7.6 universal solution algebra Vessiot extension Wronskian zero