## Preface to Advanced Algebra: Along with a companion volume Basic Algebra.

Advanced Algebra and its companion volume Basic Algebra systematically develop concepts and tools in algebra that are vital to every mathematician, whether pure or applied, aspiring or established. The two books together aim to give the reader a global view of algebra, its use, and its role in mathematics as a whole. The idea is to explain what the young mathematician needs to know about algebra in order to communicate well with colleagues in all branches of mathematics.

The books are written as textbooks, and their primary audience is students who are learning the material for the first time and who are planning a career in which they will use advanced mathematics professionally. Much of the material in the two books, including nearly all of Basic Algebra and some of Advanced Algebra, corresponds to normal course work, with the proportions depending on the university. The books include further topics that may be skipped in required courses but that the professional mathematician will ultimately want to learn by self-study. The test of each topic for inclusion is whether it is something that a plenary lecturer at a broad international or national meeting is likely to take as known by the audience. Key topics and features of Advanced Algebra are as follows:

• Topics build on the linear algebra, group theory, factorization of ideals, structure of fields, Galois theory, and elementary theory of modules developed in Basic Algebra.

• Individual chapters treat various topics in commutative and noncommutative algebra, together providing introductions to the theory of associative algebras, homological algebra, algebraic number theory, and algebraic geometry.

• The text emphasizes connections between algebra and other branches of mathematics, particularly topology and complex analysis. All the while, it carries along two themes from Basic Algebra: the analogy between integers and polynomials in one variable over a field, and the relationship between number theory and geometry.

• Several sections in two chapters introduce the subject of Grobner bases, which is the modern gateway toward handling simultaneous polynomial equations in applications.

• The development proceeds from the particular to the general, often introducing examples well before a theory that incorporates them.

**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.

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