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Undergraduate Algebra (Undergraduate Texts in Mathematics) Hardcover – March 21, 2005
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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
Top Customer Reviews
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.
This text should be required reading for all Computer Science majors who have an interest in cryptography or cryptanalysis.