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Undergraduate Algebra (Undergraduate Texts in Mathematics) Hardcover – March 21, 2005

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Editorial Reviews


From the reviews of the third edition:

"As is very typical for Professor Lang’s self demand and style of publishing, he has tried to both improve and up-date his already well-established text. … Numerous examples and exercises accompany this now already classic primer of modern algebra, which as usual, reflects the author’s great individuality just as much as his unrivalled didactic mastery and his care for profound mathematical education at any level. … The present textbook … will remain one of the great standard introductions to the subject for beginners." (Werner Kleinert, Zentralblatt MATH, Vol. 1063, 2005)

From the Back Cover

Undergraduate Algebra is a text for the standard undergraduate algebra course. It concentrates on the basic structures and results of algebra, discussing groups, rings, modules, fields, polynomials, finite fields, Galois Theory, and other topics. The author has also included a chapter on groups of matrices which is unique in a book at this level. Throughout the book, the author strikes a balance between abstraction and concrete results, which enhance each other. Illustrative examples accompany the general theory. Numerous exercises range from the computational to the theoretical, complementing results from the main text.

For the third edition, the author has included new material on product structure for matrices (e.g. the Iwasawa and polar decompositions), as well as a description of the conjugation representation of the diagonal group. He has also added material on polynomials, culminating in Noah Snyder’s proof of the Mason-Stothers polynomial abc theorem.


About the First Edition:

The exposition is down-to-earth and at the same time very smooth. The book can be covered easily in a one-year course and can be also used in a one-term course...the flavor of modern mathematics is sprinkled here and there.

- Hideyuki Matsumura, Zentralblatt


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Product Details

  • Series: Undergraduate Texts in Mathematics
  • Hardcover: 389 pages
  • Publisher: Springer; 3rd edition (March 21, 2005)
  • Language: English
  • ISBN-10: 0387220259
  • ISBN-13: 978-0387220253
  • Product Dimensions: 6.1 x 0.9 x 9.2 inches
  • Shipping Weight: 1.5 pounds (View shipping rates and policies)
  • Average Customer Review: 3.5 out of 5 stars  See all reviews (4 customer reviews)
  • Amazon Best Sellers Rank: #1,319,365 in Books (See Top 100 in Books)

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Most Helpful Customer Reviews

26 of 29 people found the following review helpful By on March 26, 1998
Format: Hardcover
I didn't enjoy this book at all. The author, as he states in the preface, has tried to strike a balance between rigour and brevity, while trying to set a standard for the American undergraduate algebra curriculum. The intentions are admirable; however, he doesn't quite pull it off. Most of the important topics are presented either in the text or in the problems. For that matter, they're well chosen, too. But quite often the presentation is either too brief, unclear, or devoid of motivation. Some definitions are ambiguous or imprecise, while the notation is sometimes clumsy. The only redeeming feature is the choice of problems. They vary in difficulty from section to section, but on an average are excellent(a factor that adds to their difficulty is that the material in the text is so skimpy). On the whole, I don't know what purpose this book serves. The average student is better off using Fraleigh or Gallian (or any of the host of easier books avalailable), while Herstein still has much to offer to the motivated one. Jacobson and Cohn(which I used a lot while taking my first Algebra course) are always available as outstanding references.
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13 of 18 people found the following review helpful By Ragnarok Books on June 26, 2004
Format: Hardcover
The book provides an introduction to abstract algebra for undergraduate students. It covers basic topics in the theory of groups, rings, fields, polynomials, vectors and modules.
The book is terse and succinct. It provides very few examples to illustrate the various definitions and theorems. The examples are sorely missing since it is probably the first encounter with abstract mathematical formalism for most students using this book.
The author does not provide motivations for most of the topics he presents. Many of the aforementioned topics are of use in physics, linear algebra and applied math, yet the author fails to mention the relevance of abstract algebra to these fields. Considering the fact that not all students using this book are pure math majors, this may leave many students asking "What is it good for?" too often.
The book contains many excellent exercises varying from the trivial to the highly challenging. However, no solutions are provided and no solutions manual is available. This makes the book highly unsuitable for self-study.
The major strength of the book is its rigor. The author covers many topics not covered by other textbooks and progresses very meticulously towards more complicated topics. He thus builds a strong foundation for future classes in abstract mathematics. For pure math students, the book is a valuable snapshot of how advanced pure math textbooks look like.
For other students looking for an introductory textbook on abstract algebra, "Abstract Algebra: An Introduction" by Thomas W. Hungerford is recommended. It provides a clearer presentation of the material and is better suitable for the beginning student.
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2 of 2 people found the following review helpful By Z. Wu on March 7, 2009
Format: Hardcover
Like all Lang books on algebra, everything is very clear. The book goes through group, ring, modules, field, vector spaces, Galois theory andsome set theory including the Axiom of Choice. It does take some work and time to get everything, especially if you are getting acquainted with the topics for the first time. However, a motivated reader can certainly work through the book by himself. The only complaint I have is that the book often does not explain the motivation much.
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6 of 12 people found the following review helpful By R. A. Williams on April 5, 2002
Format: Hardcover
The title is quite misleading. It sounds innocuous but conceals some pretty neat stuff. This book is an introduction into modern algebra that starts with group theory and continues into Galois theory. It is terse at times but contains all the essentials.
This text should be required reading for all Computer Science majors who have an interest in cryptography or cryptanalysis.
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