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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, but categories 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 Two-Squares 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 one-year course (one can also base a one-semester 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
1. Number Theory.
Introduction.
Binomial Coefficients.
Greatest Common Divisors.
The Fundamental Theorem of Arithmetic.
Congruences.
Dates and Days.
2. Groups.
Functions.
Permutations.
Groups.
Lagrage's Theorem.
Homomorphisms.
Quotient Groups.
Group Actions.
Counting with Groups.
3. Commutative Rings.
First Properties.
Fields.
Polynomials.
Greatest Common Divisors.
Unique Factorization.
Homomorphisms.
Irreducibility.
Quotient Rings and Finite Fields.
Officers, Fertilizer, and a Line at Infinity.
4. Goodies.
Linear Algebra.
Euclidean Constructions.
Classical Formulas.
Insolvability of the General Quintic.
Epilogue.
5. Modules.
Application's of Zorn's Lemma.
Modules.
Categories and Functors.
Free Modules, Projectives, and Injectives.
6. Algebras.
Noncommutative Rings.
Semisimple Rings.
Tensor Products.
Algebras.
7. Principal Ideal Domains.
8. Group Theory II.
9. Polynomials In Several Variables.
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, but categories 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 Two-Squares 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 one-year course (one can also base a one-semester 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
1. Number Theory.
Introduction.
Binomial Coefficients.
Greatest Common Divisors.
The Fundamental Theorem of Arithmetic.
Congruences.
Dates and Days.
2. Groups.
Functions.
Permutations.
Groups.
Lagrage's Theorem.
Homomorphisms.
Quotient Groups.
Group Actions.
Counting with Groups.
3. Commutative Rings.
First Properties.
Fields.
Polynomials.
Greatest Common Divisors.
Unique Factorization.
Homomorphisms.
Irreducibility.
Quotient Rings and Finite Fields.
Officers, Fertilizer, and a Line at Infinity.
4. Goodies.
Linear Algebra.
Euclidean Constructions.
Classical Formulas.
Insolvability of the General Quintic.
Epilogue.
5. Modules.
Application's of Zorn's Lemma.
Modules.
Categories and Functors.
Free Modules, Projectives, and Injectives.
6. Algebras.
Noncommutative Rings.
Semisimple Rings.
Tensor Products.
Algebras.
7. Principal Ideal Domains.
8. Group Theory II.
9. Polynomials In Several Variables.