Title: Algebraic groups and differential Galois theory /
Authors: Crespo, Teresa, 1955-; Hajto, Zbigniew.
Publisher: American Mathematical Society,
Year: 2011
Subjects: Differential algebraic groups.; Galois theory.; Morphisms (Mathematics); Field theory and polynomials -- Differential and difference algebra -- Differential algebra.; Commutative algebra -- Ring extensions and related topics -- Galois theory.; Algebraic geometry -- Foundations -- Varieties and morphisms.; Nonassociative rings and algebras -- Lie algebras and Lie superalgebras -- Lie algebras of linear algebraic groups.; Group theory and generalizations -- Linear algebraic groups and related topics -- Linear algebraic groups over arbitrary fields.; Ordinary differential equations -- Differential equations in the complex domain -- Singularities, monodromy, local behavior of solutions, normal forms.; Computer science -- Algorithms -- Symbolic computation and algebraic computation.
Abstract: "Differential Galois theory has seen intense research activity during the last decades in several directions: elaboration of more general theories, computational aspects, model theoretic approaches, applications to classical and quantum mechanics as well as to other mathematical areas such as number theory. This book intends to introduce the reader to this subject by presenting Picard-Vessiot theory, i.e. Galois theory of linear differential equations, in a self-contained way. The needed prerequisites from algebraic geometry and algebraic groups are contained in the first two parts of the book. The third part includes Picard-Vessiot extensions, the fundamental theorem of Picard-Vessiot theory, solvability by quadratures, Fuchsian equations, monodromy group and Kovacic's algorithm. Over one hundred exercises will help to assimilate the concepts and to introduce the reader to some topics beyond the scope of this book."--
Contents: Pt. 1. Algebraic geometry -- Affine and projective varieties -- Algebraic varieties -- pt. 2. Algebraic groups -- Basic notions -- Lie algebras and algebraic groups -- pt. 3. Differential Galois theory -- Picard-Vessiot extensions -- The Galois correspondence -- Differential equations over C(z) -- Suggestions for further reading.