Algebra: Pure and Applied

Algebra: Pure and Applied
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  • ISBN-13: 9780130882547
  • ISBN: 0130882542
  • Edition: 1
  • Publication Date: 2001
  • Publisher: Pearson


Aigli Papantonopoulou


This book aims to provide thorough coverage of the main topics of abstract algebra while remaining accessible to students with little or no previous exposure to abstract mathematics. It can be used either for a one-semester introductory course on groups and rings or for a full-year course. More specifics on possible course plans using the book are given in this preface. Style of Presentation Over many years of teaching abstract algebra to mixed groups of undergraduates, including mathematics majors, mathematics education majors, and computer science majors, I have become increasingly aware of the difficulties students encounter making their first acquaintance with abstract mathematics through the study of algebra. This book, based on my lecture notes, incorporates the ideas I have developed over years of teaching experience on how best to introduce students to mathematical rigor and abstraction while at the same time teaching them the basic notions and results of modern algebra. Two features of the teaching style I have found effective arerepetitionand especially anexamples first, definitions laterorder of presentation. In this book, as in my lecturing, the hard conceptual steps are always prepared for by working out concrete examples first, before taking up rigorous definitions and abstract proofs. Absorption of abstract concepts and arguments is always facilitated by first building up the student's intuition through experience with specific cases. Another principle that is adhered to consistently throughout the main body of the book (Parts A and B) is that every algebraic theorem mentioned is given either with a complete proof, or with a proof broken up into to steps that the student can easily fill in, without recourse to outside references. The book aims to provide a self-contained treatment of the main topics of algebra, introducing them in such a way that the student can follow the arguments of a proof without needing to turn to other works for help. Throughout the book all the examples, definitions, and theorems are consecutively numbered in order to make locating any particular item easier for the reader. Coverage of Topics In order to accommodate students of varying mathematical, backgrounds, an optional Chapter 0, at the beginning, collects basic material used in the development of the main theories of algebra. Included are, among other topics, equivalence relations, the binomial theorem, De Moivre's formula for complex numbers, and the fundamental theorem of arithmetic. This chapter can be included as part of an introductory course or simply referred to as needed in later chapters. Special effort is made in Chapter 1 to introduce at the beginning all main types of groups the student will be working with in later chapters. The first section of the chapter emphasizes the fact that concrete examples of groups come from different sources, such as geometry, number theory, and the theory of equations. Chapter 2 introduces the notion of group homomorphism first and then proceeds to the study of normal subgroups and quotient groups. Studying the properties of the kernel of a homomorphism before introducing the definition of a normal subgroup makes the latter notion less mysterious for the student and easier to absorb and appreciate. A similar order of exposition is adopted in connection with rings. After the basic notion of a ring is introduced in Chapter 6, Chapter 7 begins with ring homomorphisms, after which consideration of the properties of the kernels of such homomorphisms gives rise naturally to the notion of an ideal in a ring. Each chapter is designed around some central unifying theme. For instance, in Chapter 4 the concept of group action is used to unify such results as Cayley's theorem, Burnside's counting formula, the simplicy of A 5 , and the Sylow theorems and their applications. The ring of polynomials over a fieAigli Papantonopoulou is the author of 'Algebra: Pure and Applied', published 2001 under ISBN 9780130882547 and ISBN 0130882542.

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