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Zbl 0855.12001
Magid, Andy R.
Lectures on differential Galois theory. (English)
[B] University Lecture Series. 7. Providence, RI: American Mathematical Society (AMS). xiii, 104 p. \$ 35.00 (1994). ISBN 0-8218-7004-1/pbk

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. \par 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. \par Chapter 2: The Wronskian. Topics covered are the properties of the Wronskian. \par Chapter 3: Picard-Vessiot extensions. Topics covered are the definition of Picard-Vessiot extensions, their construction, and their uniqueness. \par 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. \par 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. \par 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 $SL_n$. \par 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 $GL_n$, $n\geq 3$.
[E.V.Pankrat'ev (Moskva)]

MSC 2000:: *12-02 Research monographs (field theory)
12H05 Differential algebra
12F10 Galois theory
12F20 Transcendental extensions
12F12 Inverse Galois theory

Keywords: differential ideals; automorphisms of Picard-Vessiot extensions; differential Galois theory of linear homogeneous differential equations; algebraic matrix groups; Picard-Vessiot theory; differential polynomial algebra; Wronskian; Galois correspondence; inverse Galois problem

Cited in: Zbl 1215.12001 Zbl 1097.12004 Zbl 1003.12002 Zbl 1002.12007

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