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Lectures on differential Galois theory. (English) Zbl 0855.12001

University Lecture Series. 7. Providence, RI: American Mathematical Society (AMS). xiii, 104 p. (1994).
The book deals with the differential Galois theory of linear homogeneous differential equations, whose differential Galois groups are algebraic matrix groups. This branch of the theory is known as the Picard-Vessiot theory. The book consists of seven chapters.
Chapter 1: Differential ideals. Topics include general introduction to differential polynomial algebra, characterization of ideals differentially generated by a linear homogeneous differential operator, and the fact that the quotient of the differential polynomial algebra by such an ideal is an (ordinary) polynomial ring.
Chapter 2: The Wronskian. Topics covered are the properties of the Wronskian.
Chapter 3: Picard-Vessiot extensions. Topics covered are the definition of Picard-Vessiot extensions, their construction, and their uniqueness.
Chapter 4: Automorphisms of Picard-Vessiot extensions. Topics covered are the structure of the group of automorphisms of a Picard-Vessiot extension as an algebraic group.
Chapter 5: The structure of Picard-Vessiot extensions. Topics covered include the structure of a Picard-Vessiot extension as the quotient field of an affine domain.
Chapter 6: The Galois correspondence and its consequences. Topics covered include the fundamental theorem of differential Galois theory and some applications, including equations with solvable (connected component of their) Galois group and equations solvable by quadratures, and equations with Galois group \(SL_n\).
Chapter 7: The inverse Galois problem. Topics covered include the inverse problem and derivations of the coordinate ring of an algebraic group, and the constructive solution of the inverse problem for various groups, including solvable groups and \(GL_n\), \(n\geq 3\).

MSC:

12-02 Research exposition (monographs, survey articles) pertaining to field theory
12H05 Differential algebra
12F10 Separable extensions, Galois theory
12F20 Transcendental field extensions
12F12 Inverse Galois theory
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