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More About This Textbook
Overview
This text introduces readers to the algebraic concepts of group and rings, providing a comprehensive discussion of theory as well as a significant number of applications for each.
Number Theory: Induction; Binomial Coefficients; Greatest Common Divisors; The Fundamental Theorem of Arithmetic
Congruences; Dates and Days. Groups I: Some Set Theory; Permutations; Groups; Subgroups and Lagrange's Theorem; Homomorphisms; Quotient Groups; Group Actions; Counting with Groups. Commutative Rings I: First Properties; Fields; Polynomials; Homomorphisms; Greatest Common Divisors; Unique Factorization; Irreducibility; Quotient Rings and Finite Fields; Officers, Magic, Fertilizer, and Horizons. Linear Algebra: Vector Spaces; Euclidean Constructions; Linear Transformations; Determinants; Codes; Canonical Forms. Fields: Classical Formulas; Insolvability of the General Quintic; Epilog. Groups II: Finite Abelian Groups; The Sylow Theorems; Ornamental Symmetry. Commutative Rings III: Prime Ideals and Maximal Ideals; Unique Factorization; Noetherian Rings; Varieties; Grobner Bases.
For all readers interested in abstract algebra.
Editorial Reviews
Booknews
The new edition of a textbook that introduces three related topics: number theory (division algorithm, unique factorization into primes, and congruences), group theory (permutations, Lagrange's theory, homomorphisms, and quotient groups), and commutative ring theory (domains, fields, polynomial and quotient rings, and finite fields). A final chapter combines the three topics to solve such problems as angle trisection, squaring the circle, and the construction of regular gons. Annotation c. Book News, Inc., Portland, OR (booknews.com)Product Details
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Read an Excerpt
Preface to the Second Edition
I was reluctant to accept Prentice Hall's offer to write a second edition of this book. When I wrote the first edition several years ago, I assumed the usual assumption: All first courses in algebra have essentially the same material, and so it is not necessary to ask what is in such a book, but rather how it is in it. I think that most people accept this axiom, at least tacitly, and so their books are almost all clones of one another, differing only in the quality of the writing. Looking at the first version of my book, I now see many flaws; there were some interesting ideas in it, but the book was not significantly different from others. I could improve the text I had written, but I saw no reason to redo it if I were to make only cosmetic changes.
I then thought more carefully about what an introduction to algebra ought to be. When Birkhoff and Mac Lane wrote their pioneering A Survey of Modern Algebra about 60 years ago, they chose the topics that they believed were most important, both for students with a strong interest in algebra and those with other primary interests in which algebraic ideas and methods are used. Birkhoff and Mac Lane were superb mathematicians, and they chose the topics for their book very well. Indeed, their excellent choice of topics is what has led to the clone producing assumption I have mentioned above. But times have changed; indeed, Mac Lane himself has written a version of A Survey of Modern Algebra from a categorical point of view. I feel it is too early to mention categories explicitly in this book, for I believe one learns from the particular to the general, butcategories are present implicitly in the almost routine way homomorphisms are introduced as soon as possible after introducing algebraic systems. Whereas emphasis on rings and groups is still fundamental, there are today major directions which either did not exist in 1940 or were not then recognized to be so important. These new directions involve algebraic geometry, computers, homology, and representations. One may view this new edition as the first of a two volume sequence. This book, the first volume, is designed for students beginning their study of algebra. The sequel, designed for beginning graduate students, is designed to be independent of this one. Hence, the sequel will have a substantial overlap with this book, but it will go on to discuss some of the basic results which lead to the most interesting contemporary topics. Each generation should survey algebra to make it serve the present time.
When I was writing this second edition, I was careful to keep the pace of the exposition at its original level; one should not rush at the beginning. Besides rewriting and rearranging theorems, examples, and exercises that were present in the first edition, I have added new material. For example, there is a short subsection on euclidean rings which contains a proof of Fermat's TwoSquares Theorem; and the Fundamental Theorem of Galois Theory is stated and used to prove the Fundamental Theorem of Algebra: the complex numbers are algebraically closed.
I have also added two new chapters, one with more group theory and one with more commutative rings, so that the book is now more suitable for a oneyear course (one can also base a onesemester course on the first three chapters). The new chapter on groups proves the Sylow theorems, the Jordan Holder theorem, and the fundamental theorem of finite abelian groups, and it introduces free groups and presentations by generators and relations. The new chapter on rings discusses prime and maximal ideals, unique factorization in polynomial rings in several variables, noetherian rings, varieties, and Grobner bases. Finally, a new section contains hints for most of the exercises (and an instructor's solution manual contains complete solutions for all the exercises in the first four chapters).
In addition to thanking again those who helped me with the first edition, it is a pleasure to thank Daniel D. Anderson, Aldo Brigaglia, E. Graham Evans, Daniel Flath, William Haboush, Dan Grayson, Christopher Heil, Gerald J. Janusz, Jennifer D. Key, Steven L. Kleiman, Emma Previato, Juan Jorge Schaffer, and Thomas M. Songer for their valuable suggestions for this book.
And so here is edition two; my hope is that it makes modern algebra accessible to beginners, and that it will make its readers want to pursue algebra further.
Joseph J. Rotman
Table of Contents
Induction
Binomial Coefficients
Greatest Common Divisors
The Fundamental Theorem of Arithmetic
Congruences
Dates and Days
Chapter 2: Groups I
Some Set Theory
Permutations
Groups
Subgroups and Lagrange's Theorem
Homomorphisms
Quotient Groups
Group Actions
Counting with Groups
Chapter 3: Commutative Rings I
First Properties
Fields
Polynomials
Homomorphisms
Greatest Common Divisors
Unique Factorization
Irreducibility
Quotient Rings and Finite Fields
Officers, Magic, Fertilizer, and Horizons
Chapter 4: Linear Algebra
Vector Spaces
Euclidean Constructions
Linear Transformations
Determinants
Codes
Canonical Forms
Chapter 5: Fields
Classical Formulas
Insolvability of the General Quintic
Epilog
Chapter 6: Groups II
Finite Abelian Groups
The Sylow Theorems
Ornamental Symmetry
Chapter 7: Commutative Rings III
Prime Ideals and Maximal Ideals
Unique Factorization
Noetherian Rings
Varieties
Grobner Bases
Hints for Selected Exercises
Bibliography
Index
Preface
Preface to the Second Edition
I was reluctant to accept Prentice Hall's offer to write a second edition of this book. When I wrote the first edition several years ago, I assumed the usual assumption: All first courses in algebra have essentially the same material, and so it is not necessary to ask what is in such a book, but rather how it is in it. I think that most people accept this axiom, at least tacitly, and so their books are almost all clones of one another, differing only in the quality of the writing. Looking at the first version of my book, I now see many flaws; there were some interesting ideas in it, but the book was not significantly different from others. I could improve the text I had written, but I saw no reason to redo it if I were to make only cosmetic changes.
I then thought more carefully about what an introduction to algebra ought to be. When Birkhoff and Mac Lane wrote their pioneering A Survey of Modern Algebra about 60 years ago, they chose the topics that they believed were most important, both for students with a strong interest in algebra and those with other primary interests in which algebraic ideas and methods are used. Birkhoff and Mac Lane were superb mathematicians, and they chose the topics for their book very well. Indeed, their excellent choice of topics is what has led to the clone producing assumption I have mentioned above. But times have changed; indeed, Mac Lane himself has written a version of A Survey of Modern Algebra from a categorical point of view. I feel it is too early to mention categories explicitly in this book, for I believe one learns from the particular to the general,butcategories are present implicitly in the almost routine way homomorphisms are introduced as soon as possible after introducing algebraic systems. Whereas emphasis on rings and groups is still fundamental, there are today major directions which either did not exist in 1940 or were not then recognized to be so important. These new directions involve algebraic geometry, computers, homology, and representations. One may view this new edition as the first of a two volume sequence. This book, the first volume, is designed for students beginning their study of algebra. The sequel, designed for beginning graduate students, is designed to be independent of this one. Hence, the sequel will have a substantial overlap with this book, but it will go on to discuss some of the basic results which lead to the most interesting contemporary topics. Each generation should survey algebra to make it serve the present time.
When I was writing this second edition, I was careful to keep the pace of the exposition at its original level; one should not rush at the beginning. Besides rewriting and rearranging theorems, examples, and exercises that were present in the first edition, I have added new material. For example, there is a short subsection on euclidean rings which contains a proof of Fermat's TwoSquares Theorem; and the Fundamental Theorem of Galois Theory is stated and used to prove the Fundamental Theorem of Algebra: the complex numbers are algebraically closed.
I have also added two new chapters, one with more group theory and one with more commutative rings, so that the book is now more suitable for a oneyear course (one can also base a onesemester course on the first three chapters). The new chapter on groups proves the Sylow theorems, the Jordan Holder theorem, and the fundamental theorem of finite abelian groups, and it introduces free groups and presentations by generators and relations. The new chapter on rings discusses prime and maximal ideals, unique factorization in polynomial rings in several variables, noetherian rings, varieties, and Grobner bases. Finally, a new section contains hints for most of the exercises (and an instructor's solution manual contains complete solutions for all the exercises in the first four chapters).
In addition to thanking again those who helped me with the first edition, it is a pleasure to thank Daniel D. Anderson, Aldo Brigaglia, E. Graham Evans, Daniel Flath, William Haboush, Dan Grayson, Christopher Heil, Gerald J. Janusz, Jennifer D. Key, Steven L. Kleiman, Emma Previato, Juan Jorge Schaffer, and Thomas M. Songer for their valuable suggestions for this book.
And so here is edition two; my hope is that it makes modern algebra accessible to beginners, and that it will make its readers want to pursue algebra further.
Joseph J. Rotman