Advanced Algebra - Along with a Companion Volume 'Basic Algebra'

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. Advanced Algebra includes chapters on modern algebra which treat various topics in commutative and noncommutative algebra and provide introductions to the theory of associative algebras, homological algebras, algebraic number theory, and algebraic geometry. Together the two books give the reader a global view of algebra, its role in mathematics as a whole and are suitable as texts in a two-semester advanced undergraduate or first-year graduate sequence in algebra.

List of contents

Transition to Modern Number Theory.- Wedderburn-Artin Ring Theory.- Brauer Group.- Homological Algebra.- Three Theorems in Algebraic Number Theory.- Reinterpretation with Adeles and Ideles.- Infinite Field Extensions.- Background for Algebraic Geometry.- The Number Theory of Algebraic Curves.- Methods of Algebraic Geometry.

Summary

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. Advanced Algebra includes chapters on modern algebra which treat various topics in commutative and noncommutative algebra and provide introductions to the theory of associative algebras, homological algebras, algebraic number theory, and algebraic geometry. Together the two books give the reader a global view of algebra, its role in mathematics as a whole and are suitable as texts in a two-semester advanced undergraduate or first-year graduate sequence in algebra.

Additional text

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)

Report

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)

Product details

• Authors: Anthony W Knapp, Anthony W. Knapp
• Publisher: Springer, Basel
• Languages: English
• Product format: Hardback
• Released: 04.04.2008
• EAN: 9780817645229
• ISBN: 978-0-8176-4522-9
• No. of pages: 730
• Dimensions: 157 mm x 48 mm x 245 mm
• Weight: 1335 g
• Illustrations: XXV, 730 p. 46 illus.
• Series: Cornerstones; Cornerstones
• Subjects: Natural sciences, medicine, IT, technology > Mathematics > Arithmetic, algebra; Algebra, B, Mathematics and Statistics, Algebraic Geometry, Number Theory, Rings (Algebra), Non-associative Rings and Algebras, Nonassociative rings, Mathematical foundations, Field Theory and Polynomials, Field theory (Physics), Category theory (Mathematics), Category Theory, Homological Algebra, Homological algebra