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This carefully written textbook offers a thorough introduction to abstract algebra, covering the fundamentals of groups, rings and fields.
The first two chapters present preliminary topics such as properties of the integers and equivalence relations. The author then explores the first major algebraic structure, the group, progressing as far as the Sylow theorems and the classification of finite abelian groups. An introduction to ring theory follows, leading to a discussion of fields and polynomials that includes sections on splitting fields and the construction of finite fields. The final part contains applications to public key cryptography as well as classical straightedge and compass constructions.
Explaining key topics at a gentle pace, this book is aimed at undergraduate students. It assumes no prior knowledge of the subject and contains over 500 exercises, half of which have detailed solutions provided.
List of contents
Part I Preliminaries.- 1 Relations and Functions.- 2 The Integers and Modular Arithmetic.- Part II Groups.- 3 Introduction to Groups.- 4 Factor Groups and Homomorphisms.- 5 Direct Products and the Classification of Finite Abelian Groups.- 6 Symmetric and Alternating Groups.- 7 The Sylow Theorems.- Part III Rings.- 8 Introduction to Rings.- 9 Ideals, Factor Rings and Homomorphisms.- 10 Special Types of Domains.- Part IV Fields and Polynomials.- 11 Irreducible Polynomials.- 12 Vector Spaces and Field Extensions.- Part V Applications.- 13 Public Key Cryptography.- 14 Straightedge and Compass Constructions.- A The Complex Numbers.- B Matrix Algebra.- Solutions.- Index.
About the author
Gregory T. Lee is a professor at Lakehead University specializing in group rings, a branch of abstract algebra. He has published numerous papers on the subject, as well as a monograph with Springer.
Summary
This carefully written textbook offers a thorough introduction to abstract algebra, covering the fundamentals of groups, rings and fields.
The first two chapters present preliminary topics such as properties of the integers and equivalence relations. The author then explores the first major algebraic structure, the group, progressing as far as the Sylow theorems and the classification of finite abelian groups. An introduction to ring theory follows, leading to a discussion of fields and polynomials that includes sections on splitting fields and the construction of finite fields. The final part contains applications to public key cryptography as well as classical straightedge and compass constructions.
Explaining key topics at a gentle pace, this book is aimed at undergraduate students. It assumes no prior knowledge of the subject and contains over 500 exercises, half of which have detailed solutions provided.
Additional text
“The book is very clearly written. The author successfully presents the material in an appealing way. A big number of examples enriches the text and enlightens the key topics. Exercises of different level are included at the end of each chapter and solutions to approximately half of the exercises are included at the very end of the book. In summary … the book can definitely be recommended as text book for a first introduction to abstract algebra.” (C. Fuchs, Internationale Mathematische Nachrichten IMN, Vol. 73 (240), April, 2019)
“The book provides the reader with valuable technical information regarding the introductory notions and main results of abstract algebra. The author presents concepts, theorems and applications in a very clear and fluent way within the manuscript. Thus, ‘Abstract Algebra. An Introductory Course’ is obviously a well written document with respect to the field of abstract algebra.” (Diana Maimut, zbMATH 1401.00003, 2019)
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"The book is very clearly written. The author successfully presents the material in an appealing way. A big number of examples enriches the text and enlightens the key topics. Exercises of different level are included at the end of each chapter and solutions to approximately half of the exercises are included at the very end of the book. In summary ... the book can definitely be recommended as text book for a first introduction to abstract algebra." (C. Fuchs, Internationale Mathematische Nachrichten IMN, Vol. 73 (240), April, 2019)
"The book provides the reader with valuable technical information regarding the introductory notions and main results of abstract algebra. The author presents concepts, theorems and applications in a very clear and fluent way within the manuscript. Thus, 'Abstract Algebra. An Introductory Course' is obviously a well written document with respect to the field of abstract algebra." (Diana Maimut, zbMATH 1401.00003, 2019)
