© 2016

Algebra I

Textbook for Students of Mathematics

  • Challenging amount of material thoughtfully organized for deep and fast learning

  • Large collection of exercises equipped with hints and a lot of problems for independent solution

  • Simple modern explanation of subjects usually omitted in basic courses, such as representations of the symmetric group, geometry of algebraic varieties, meaty aspects of the category theory etc


Table of contents

  1. Front Matter
    Pages i-xx
  2. Alexey L. Gorodentsev
    Pages 1-18
  3. Alexey L. Gorodentsev
    Pages 19-40
  4. Alexey L. Gorodentsev
    Pages 41-71
  5. Alexey L. Gorodentsev
    Pages 73-102
  6. Alexey L. Gorodentsev
    Pages 103-122
  7. Alexey L. Gorodentsev
    Pages 123-154
  8. Alexey L. Gorodentsev
    Pages 155-171
  9. Alexey L. Gorodentsev
    Pages 173-204
  10. Alexey L. Gorodentsev
    Pages 205-228
  11. Alexey L. Gorodentsev
    Pages 229-252
  12. Alexey L. Gorodentsev
    Pages 253-278
  13. Alexey L. Gorodentsev
    Pages 279-307
  14. Alexey L. Gorodentsev
    Pages 309-334
  15. Alexey L. Gorodentsev
    Pages 335-360
  16. Alexey L. Gorodentsev
    Pages 361-386
  17. Alexey L. Gorodentsev
    Pages 387-420
  18. Alexey L. Gorodentsev
    Pages 421-457
  19. Alexey L. Gorodentsev
    Pages 459-479
  20. Alexey L. Gorodentsev
    Pages 481-502

About this book


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.


Fields Rings Modules Groups Linear Algebra Multilinear Algebra Representation Theory Commutative Algebra Algebraic Varieties Galois Theory

Authors and affiliations

  1. 1.Faculty of MathematicsNational Research University “Higher School of Economics”MoscowRussia

About the authors

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.

Bibliographic information

  • Book Title Algebra I
  • Book Subtitle Textbook for Students of Mathematics
  • Authors Alexey L. Gorodentsev
  • DOI
  • Copyright Information Springer International Publishing AG 2016
  • Publisher Name Springer, Cham
  • eBook Packages Mathematics and Statistics Mathematics and Statistics (R0)
  • Hardcover ISBN 978-3-319-45284-5
  • Softcover ISBN 978-3-319-83257-9
  • eBook ISBN 978-3-319-45285-2
  • Edition Number 1
  • Number of Pages XX, 564
  • Number of Illustrations 37 b/w illustrations, 42 illustrations in colour
  • Topics Algebra
  • Buy this book on publisher's site


“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)