Algebra

From April 1999 Notices of the AMS, announcing that the author was awarded the Leroy P. Steele Prize for Mathematical Exposition for his many mathematics books: “Lang’s Algebra changed the way graduate algebra is taught, retaining classical topics but introducing language and ways of thinking from category theory and homological algebra. It has affected all subsequent graduate-level algebra books.”

From MathSciNet’s review of the first edition:

“The author has an impressive knack for presenting the important and interesting ideas of algebra in just the “right” way, and he never gets bogged down in the dry formalism which pervades some parts of algebra.”

This book is intended as a basic text for a one-year course in Algebra at the graduate level, or as a useful reference for mathematicians and professionals who use higher-level algebra. This book successfully addresses all of the basic concepts of algebra. For the new edition, the author has added exercises and made numerous corrections to the text.

Table of Contents

Part 1: The Basic Objects of Algebra

Chapter 1: Groups

Chapter 2: Rings

Chapter 3: Modules

Chapter 4: Polynomials

Part 2: Algebraic Equations

Chapter 5: Algebraic Extensions

Chapter 6: Galois Theory

Chapter 7: Extensions of Rings

Chapter 8: Transcendental Extensions

Chapter 9: Algebraic Spaces

Chapter 10: Noetherian Rings and Modules

Chapter 11: Real Fields

Chapter 12: Absolute Values

Part 3: Linear Algebra and Representations

Chapter 13: Matrices and Linear Maps

Chapter 14: Representation of One Endomorphism

Chapter 15: Structure of Bilinear Forms

Chapter 16: The Tensor Product

Chapter 17: Semisimplicity

Chapter 18: Representations of Finite Groups

Chapter 19: The Alternating Product

Part 4: Homological Algebra

Chapter 20: General Homology Theory

Chapter 21: Finite Free Resolutions

Review

“Lang’s Algebra changed the way graduate algebra is taught, retaining classical topics but introducing language and ways of thinking from category theory and homological algebra. It has affected all subsequent graduate-level algebra books.”―NOTICES OF THE AMS

“The author has an impressive knack for presenting the important and interesting ideas of algebra in just the ‘right’ way, and he never gets bogged down in the dry formalism which pervades some parts of algebra.”―MATHEMATICAL REVIEWS

From the reviews of the third edition:

“The current third edition has grown again … dealing with topics close to the author’s heart from number theory, function theory and algebraic geometry. For the math graduate who wants to broaden his education this is an excellent account; apart from standard topics it picks out many items from other fields … . This makes it a fascinating book to read … . a very readable treatment of many modern mainline topics as well as some interesting out-of-the-way items.” (Paul M. Cohn, Zentralblatt MATH, Vol. 984, 2003)

“Lang’s Algebra … has gained an iconic status, due both to the comprehensiveness of its coverage and its ability to be authoritative and lively at the same time. … a revolutionary work, changing the way in which graduate algebra was taught. … the author describes the book as ‘very stable’, indicating that there is little that he has wished to change. This confidence is reflected in the wider mathematical community, and … this new printing deserves a place in every university departmental library.” (Gerry Leversha, The Mathematical Gazette, Vol. 87 (509), 2003)