Algebra II: Textbook for Students of Mathematics

Description

This book is the second volume of an intensive “Russian-style” two-year undergraduate course in abstract algebra, and introduces readers to the basic algebraic structures – fields, rings, modules, algebras, groups, and categories – and explains the main principles of and methods for working with them. The course covers substantial areas of advanced combinatorics, geometry, linear and multilinear algebra, representation theory, category theory, commutative algebra, Galois theory, and algebraic geometry – topics that are often overlooked in standard undergraduate courses. This textbook is based on courses the author has conducted at the Independent University of Moscow and at the Faculty of Mathematics in the Higher School of Economics. The main content is complemented by a wealth of exercises for class discussion, some of which include comments and hints, as well as problems for independent study. Front Matter....Pages i-xv Tensor Products....Pages 1-19 Tensor Algebras....Pages 21-56 Symmetric Functions....Pages 57-73 Calculus of Arrays, Tableaux, and Diagrams....Pages 75-98 Basic Notions of Representation Theory....Pages 99-129 Representations of Finite Groups in Greater Detail....Pages 131-150 Representations of Symmetric Groups....Pages 151-171 \(\mathfrak{s}\mathfrak{l}_{2}\) -Modules....Pages 173-186 Categories and Functors....Pages 187-226 Extensions of Commutative Rings....Pages 227-240 Affine Algebraic Geometry....Pages 241-263 Algebraic Manifolds....Pages 265-293 Algebraic Field Extensions....Pages 295-314 Examples of Galois Groups....Pages 315-334 Back Matter....Pages 335-370