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This book is the first volume of an intensive "Russian-style" two-year graduate 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.
List of contents
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.
About the author
A.L. Gorodentsev is professor at the Independent University of Moscow and at the Faculty of Mathematics at the National Research University "Higher School of Economics".
He is working in the field of algebraic and symplectic geometry, homological algebra and representation theory connected with geometry of algebraic and symplectic varieties.
He is one of the first developers of the "Helix Theory" and semiorthogonal decomposition technique for studying the derived categories of coherent sheaves.
Summary
This book is the first volume of an intensive “Russian-style” two-year graduate 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.
Report
"The book weaves together a great deal of classical material with a modern approach. ... the book consists of the large number of exercises ... which the author assigns for homework and the problems for independent study provided at the end of each chapter; a student who works through these will be well rewarded. ... instructors would find it more suitable for a graduate course in which the students are already familiar with the more elementary parts of the material." (John D. Dixon, zbMATH 1359.15001, 2017)
