Authors:
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 selfstudy
Part of the book series: Cornerstones (COR)
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Table of contents (10 chapters)

Front Matter

Back Matter
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 forwardlooking way that takes into account the historical development of the subject. It is suitable as a text for the more advanced parts of a twosemester firstyear 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 wellwritten book, characterized by clear and ‘userfriendly’ 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
Book Title: Advanced Algebra
Authors: Anthony W. Knapp
Series Title: Cornerstones
DOI: https://doi.org/10.1007/9780817646134
Publisher: Birkhäuser Boston, MA
eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)
Copyright Information: Birkhäuser Boston 2008
Hardcover ISBN: 9780817645229
eBook ISBN: 9780817646134
Series ISSN: 2197182X
Series EISSN: 21971838
Edition Number: 1
Number of Pages: XXV, 730
Number of Illustrations: 46 b/w illustrations
Topics: Algebra, Nonassociative Rings and Algebras, Field Theory and Polynomials, Algebraic Geometry, Number Theory, Category Theory, Homological Algebra