**University Lecture Series**

Volume: 7;
1994;
105 pp;
Softcover

MSC: Primary 12;

Print ISBN: 978-0-8218-7004-4

Product Code: ULECT/7

List Price: $25.00

AMS Member Price: $20.00

MAA Member Price: $22.50

**Electronic ISBN: 978-1-4704-2156-4
Product Code: ULECT/7.E**

List Price: $25.00

AMS Member Price: $20.00

MAA Member Price: $22.50

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# Lectures on Differential Galois Theory

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*Andy R. Magid*

Differential 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 solutions 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.

#### Readership

Graduate students and researchers.

#### Reviews & Endorsements

The present book offers an elegant alternative approach to the Galois theory of linear homogeneous differential equations, based on the principle that the Galois correspondence should be obtained as a consequence of the algebraic group-theoretic construction of Picard-Vessiot extensions.

-- Mathematical Reviews

The self-contained introduction Magid's 100-page book provides should help the newcomer to proceed further into this beautiful and active field.

-- Bulletin of the AMS

#### Table of Contents

# Table of Contents

## Lectures on Differential Galois Theory

- Cover Cover11 free
- Title iii4 free
- Copyright iv5 free
- Contents v6 free
- Preface vii8 free
- Outline of approach viii9 free
- Chapter-by-chapter summary xiii14 free
- CHAPTER 1. Differential Ideals 116 free
- CHAPTER 2. The Wronskian 1530
- CHAPTER 3. Picard–Vessiot Extensions 2338
- CHAPTER 4. Automorphisms of Picard–Vessiot Extensions 4358
- CHAPTER 5. The Structure of Picard–Vessiot Extensions 6176
- CHAPTER 6. The Galois Correspondence and its Consequences 7590
- CHAPTER 7. The Inverse Galois Problem 89104
- Bibliography 103118
- Back Cover Back Cover1119