"[The first] ten chapters...are an efficient, accessible, and self-contained introduction to affine algebraic groups over an algebraically closed field. The author includes exercises and the book is certainly usable by graduate students as a text or for self-study...the author [has a] student-friendly style... [The following] seven chapters... would also be a good introduction to rationality issues for algebraic groups. A number of results from the literature...appear for the first time in a text."

-Mathematical Reviews (Review of the Second Edition)

Klappentext

The first edition of this book presented the theory of linear algebraic groups over an algebraically closed field. The second edition, thoroughly revised and expanded, extends the theory over arbitrary fields, which are not necessarily algebraically closed. It thus represents a higher aim. As in the first edition, the book includes a self-contained treatment of the prerequisites from algebraic geometry and commutative algebra, as well as basic results on reductive groups. As a result, the first part of the book can well serve as a text for an introductory graduate course on linear algebraic groups.

Inhalt

Some Algebraic Geometry.- Linear Algebraic Groups, First Properties.- Commutative Algebraic Groups.- Derivations, Differentials, Lie Algebras.- Topological Properties of Morphisms, Applications.- Parabolic Subgroups, Borel Subgroups, Solvable Groups.- Weyl Group, Roots, Root Datum.- Reductive Groups.- The Isomorphism Theorem.- The Existence Theorem.- More Algebraic Geometry.- F-groups: General Results.- F-tori.- Solvable F-groups.- Freductive Groups.- Reductive F-Groups.- Classification.