Advanced Algebra
Anthony W. Knapp
Birkhäuser Boston
<P><EM>Basic Algebra</EM> and <EM>Advanced Algebra</EM> 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.</P>  <P>Key topics and features of <EM>Advanced Algebra</EM>:</P>  <P>*Topics build upon the linear algebra, group theory, factorization of ideals, structure of fields, Galois theory, and elementary theory of modules as developed in <EM>Basic Algebra</EM></P>  <P></P>  <P>*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</P>  <P></P>  <P>*Sections in two chapters relate the theory to the subject of Gröbner bases, the foundation for handling systems of polynomial equations in computer applications</P>  <P></P>  <P>*Text emphasizes connections between algebra and other branches of mathematics, particularly topology and complex analysis</P>  <P>*Book carries on two prominent themes recurring in <EM>Basic Algebra</EM>: the analogy between integers and polynomials in one variable over a field, and the relationship between number theory and geometry</P>  <P>*Many examples and hundreds of problems are included, along with hints or complete solutions for most of the problems</P>  <P>*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</P>  <P><EM>Advanced Algebra</EM> 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 <EM>Basic Algebra</EM>.