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  • Textbook
  • © 2008

Advanced Algebra

Birkhäuser
  • Focuses on algebraic number theory and algebraic geometry, and the distinctions and influences of each topic on the other

  • Presents material vital to the knowledge of the professional mathematician

  • Comprehensive text ideal for the classroom or self-study

Part of the book series: Cornerstones (COR)

Buying options

eBook USD 119.00
Price excludes VAT (USA)
  • ISBN: 978-0-8176-4613-4
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
Hardcover Book USD 159.99
Price excludes VAT (USA)

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Table of contents (10 chapters)

  1. Front Matter

    Pages i-xxiv
  2. Transition to Modern Number Theory

    • Anthony W. Knapp
    Pages 1-75
  3. Wedderburn–Artin Ring Theory

    • Anthony W. Knapp
    Pages 76-122
  4. Brauer Group

    • Anthony W. Knapp
    Pages 123-165
  5. Homological Algebra

    • Anthony W. Knapp
    Pages 166-261
  6. Three Theorems in Algebraic Number Theory

    • Anthony W. Knapp
    Pages 262-312
  7. Reinterpretation with Adeles and Ideles

    • Anthony W. Knapp
    Pages 313-402
  8. Infinite Field Extensions

    • Anthony W. Knapp
    Pages 403-446
  9. Background for Algebraic Geometry

    • Anthony W. Knapp
    Pages 447-519
  10. The Number Theory of Algebraic Curves

    • Anthony W. Knapp
    Pages 520-557
  11. Methods of Algebraic Geometry

    • Anthony W. Knapp
    Pages 558-648
  12. Back Matter

    Pages 649-730

About this book

Basic Algebra and Advanced Algebra systematically develop concepts and tools in algebra that are vital to every mathematician, whether pure or applied, aspiring or established. Together, the two books give the reader a global view of algebra and its role in mathematics as a whole.

Key topics and features of Advanced Algebra:

*Topics build upon the linear algebra, group theory, factorization of ideals, structure of fields, Galois theory, and elementary theory of modules as developed in Basic Algebra

*Chapters treat various topics in commutative and noncommutative algebra, providing introductions to the theory of associative algebras, homological algebra, algebraic number theory, and algebraic geometry

*Sections in two chapters relate the theory to the subject of Gröbner bases, the foundation for handling systems of polynomial equations in computer applications

*Text emphasizes connections between algebra and other branches of mathematics, particularly topology and complex analysis

*Book carries on two prominent themes recurring in Basic Algebra: the analogy between integers and polynomials in one variable over a field, and the relationship between number theory and geometry

*Many examples and hundreds of problems are included, along with hints or complete solutions for most of the problems

*The exposition proceeds from the particular to the general, often providing examples well before a theory that incorporates them; it includes blocks of problems that illuminate aspects of the text and introduce additional topics

Advanced Algebra presents its subject matter in a forward-looking way that takes into account the historical development of the subject. It is suitable as a text for the more advanced parts of a two-semester first-year graduate sequence in algebra. It requires of the reader only a familiarity with the topics developed in Basic Algebra.

Keywords

  • Algebra
  • Group theory
  • Homological algebra
  • algebraic geometry
  • algebraic number theory
  • finite field
  • number theory
  • ring theory

Reviews

From the reviews:

"This textbook is a sequel to the author's textbook Basic Algebra...which is an excellent introduction to groups, linear algebra, commutative rings, and Galois theory. The text under review contains the basic theory of noncommutative rings, and delves quite deeply into algebraic number theory and algebraic geometry. This reviewer finds the author's writing style extremely engaging, and shares his propensity for aiming whenever possible at an interesting and important theorem which illustrates the theory which the chapter develops...This is a beautiful book, which should serve well as a basic graduate textbook in algebra."   —Mathematical Reviews

"All together, this is another outstanding textbook written by the renowned and versatile mathematical researcher, teacher, and author Anthony W. Knapp that reflects his spirit, his devotion to mathematics, and his rich experiences in expository writing at best...This textbook is the second volume of Anthony W. Knapp's comprehensive introduction to the fundamental concepts and tools in modern abstract algebra. Together with its foregoing companion volume Basic Algebra, which was published in the autumn of 2006, the current book is to provide a global view of the subject, thereby particularly emphasizing both its various applications and its ubiquitous role in contemporary mathematics. As the author already pointed out in the preface to the first volume, his leading idea was to give a systematic account of what a budding mathematician needs to know about the principles of modern algebra in order to communicate well with colleagues in all branches of mathematics and related sciences. This rewarding program was masterly begun in the companion volume Basic Algebra, where the fundamentals of linear algebra, multilinear algebra, group theory, commutative algebra, field theory, Galois theory, and module theory over noncommutative rings were profoundly developed."   —Zentralblatt Math

"Advanced Algebra is a wonderfully useful and well-written book, characterized by clear and ‘user-friendly’ treatments of many important algebraic topics. … Finally, Advanced Algebra contains a thorough coverage of Gröbner bases … and continues the trend set in Basic Algebra of providing good, meaningful (and plentiful) exercises. I highly recommend this wonderful book." (Michael Berg, MAA Online, January, 2008)

Authors and Affiliations

  • East Setauket, USA

    Anthony W. Knapp

Bibliographic Information

Buying options

eBook USD 119.00
Price excludes VAT (USA)
  • ISBN: 978-0-8176-4613-4
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
Hardcover Book USD 159.99
Price excludes VAT (USA)