A First Course in Abstract Algebra: Rings, Groups and Fields, second edition

This is an interesting undergraduate textbook on general algebra. It begins with algebraic and number theory properties of integers and of polynomials over rational numbers and it develops the standard theory of rings, fields and groups, ending with the basics of Galois theory. It contains the following chapters: I. Numbers, Polynomials and Factoring; II. Rings, Domains and Fields; III. Unique Factorization; IV. Ring homomorphisms and Ideals; V. Groups; VI. Group Homomorphisms and Permutations; VII. Constructibility Problems; VIII. Vector Spaces and Field Extensions; IX. Galois Theory. A remarkable feature of the book is that it starts with the concept of a ring, only introducing groups later. The reason for this is that students are usually more familiar with various number domains than with mappings and matrices. There are a huge number of examples in this book; abstract notions are built on these examples and motivated by historical remarks explaining the development of the abstract approach to modern algebra. The book contains a lot of nice exercises of varying degrees of difficulty so it can also be used as a practice book. Last but not least, there are two interesting chapters with classical applications of modern algebra: the impossibility of certain constructions with straightedge and compass, and (non-)solvability of polynomials in radicals.

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