Algebra I [electronic resource] : Textbook for Students of Mathematics / by Alexey L. Gorodentsev.

Notations and Abbreviations -- 1 Set-Theoretic and Combinatorial Background -- 2 Integers and Residues -- 3 Polynomials and Simple Field Extensions -- 4 Elementary Functions and Power Series Expansions -- 5 Ideals, Quotient Rings, and Factorization -- 6 Vectors -- 7 Duality -- 8 Matrices -- 9 Determinants -- 10 Euclidean Spaces -- 11 Projective Spaces -- 12 Groups -- 13 Description of Groups -- 14 Modules over a Principal Ideal Domain -- 15 Linear Operators -- 16 Bilinear Forms -- 17 Quadratic Forms and Quadrics -- 18 Real Versus Complex -- 19 Hermitian Spaces -- 20 Quaternions and Spinors -- References -- Hints to Selected Exercises -- Index.

This book is the first 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.