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.

Review

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)

Table of Contents

I. TRANSITION TO MODERN NUMBER THEORY

II. WEDDERBURN–ARTIN RING THEORY

III. BRAUER GROUP

IV. HOMOLOGICAL ALGEBRA

V. THREE THEOREMS IN ALGEBRAIC NUMBER THEORY

VI. REINTERPRETATIONWITH ADELES AND IDELES

VII. INFINITE FIELD EXTENSIONS

VIII. BACKGROUND FOR ALGEBRAIC GEOMETRY

IX. THE NUMBER THEORY OF ALGEBRAIC CURVES

X. METHODS OF ALGEBRAIC GEOMETRY

X. METHODS OF ALGEBRAIC GEOMETRY (Continued)

About the Author

Anthony W. Knapp is a mathematician and professor known for his work in pure mathematics, particularly in the areas of Lie groups, representation theory, and harmonic analysis. Across his career, he has contributed to the understanding of the structure and representations of Lie groups, explored topics related to induced representations and analysis on manifolds, and produced clear expository writing that helps students grasp advanced concepts in Lie theory. He has been affiliated with various universities, engaging in research, teaching, and mentorship, and his scholarly influence is reflected in his publications and potential textbooks or monographs in the field.