Basic Algebra

From the reviews: Finally, the author's notorious masterly style of writing, which stands out by its high degree of clarity, elegance, refinement, and accuracy, also rules over this newest textbook of his, which is very likely to become one of the great standard texts in algebra for generations. -Zentralblatt Math "Anthony W. Knapp Basic Algebra aims to teach the student algebra in a somewhat different way than is the norm. Together with its sequel, Advanced Algebra, the book proposes to take an aspiring mathematician from his first exposure to algebra proper, right after calculus, through his graduate training in algebra -- and a bit further... "To use Basic Algebra in a standard undergraduate curriculum would entail spreading the material over several semesters, since the book can be used for linear algebra, group theory (Chapter IV is a gem!), rings and fields, Galois theory, a second course in group theory, and even a course on modules. "Basic Algebra is a very interesting and well-written book, and is indeed well suited for the approach to algebra the author intends, and, for that matter, usable for commonplace approaches as well." -MAA Reviews "This is a basic textbook in abstract algebra suitable for a first-year graduate algebra course or perhaps a rather challenging upper level undergraduate course. ... The appendices contain some basic set theory. ... There are many interesting problems, in which a considerable amount of additional material is developed. In particular the theme of coding theory is pursued through many of the chapter exercises and topics such as MacWilliams identity and BCH codes are broached." (Ronald Solomon, Mathematical Reviews, Issue 2007 e) "This book is written as a textbook for students who are learning algebra for the first time. Much of the material corresponds to the contents of a usual algebra course. ... numerous further topics are included which go beyond the level of an undergraduate or even first year graduate student." (H. Mitsch, Monatshefte fur Mathematik, Vol. 156 (4), April, 2009)

• Preliminaries about the Integers, Polynomials, and Matrices.- Vector Spaces over ?, ?, and ?.- Inner-Product Spaces.- Groups and Group Actions.- Theory of a Single Linear Transformation.- Multilinear Algebra.- Advanced Group Theory.- Commutative Rings and Their Modules.- Fields and Galois Theory.- Modules over Noncommutative Rings.