Advanced 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. 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.

Klappentext

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. Many examples and hundreds of problems are included, along with hints or complete solutions for most of the problems. Together the two books give the reader a global view of algebra and its role in mathematics as a whole.

Inhalt

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.