When reviewing a book on modern algebra the issue is not only how good the book good is, but also for whom it is good. This book is an excellent book for an upper-level, undergraduate, one or two semester course, in modern algebra, for a typical University student population that is not especially strong in proofs.
What I particularly like about the book is the following.
Doable Exercises: The strongest point of the book is the richness and diverse flavor of over 1000 exercises. There are proof exercises but most exercises are non-routine computations or verifications. The issue with general proof exercises is that weak students can attempt them and get nowhere, thus wasting time and encouraging them to give up. An exercise that has thinking aspects but is based on non-routine computation or verification can be done with enough work even by a weak student. This stimulates and motivates.
An illustration of the computational-verification flavor of the exercises is afforded by the first section in the chapter on group theory which has 37 exercises.
Exercise Richness: The text excels in both quantity and quality of exercises. The exercises have a rich diversity of color as the following examples illustrate.
History: The author wrote this book from the historical point of view. This can indeed be exciting to a student interested in what mathematics is like. I myself found it interesting to see original excerpts from the masters such as al-Khwarizmi (solution of the quadratic equation), Cardano (solution of the cubic), Abel (unsolvability of the quintic), Galois (foundations of Galois theory) and of course Cayley (enumeration of groups by looking at permutation groups). I believe the real strength in using a historical approach is the wealth of computational examples it invites. This is felt throughout the book where exercises challenge students to apply the theory to solve equations of degree 3 or 4 over the complex and finite fields as well as factorizations over rings over the integers.
Modern Look: The book has all the characteristics of many modern textbooks: a) accompanying diagrams, b) adequate illustrative examples in each section, c) chapter summaries, d) a list of new terms at the end of each chapter, e) chapter review exercises, f) supplementary chapter exercises, g) solutions to odd number exercises, h) appendices covering induction and logic in adequate depth, i) a modest bibliography and a j) neat collection of one-paragraph biographies of about two dozen mathematicians.
Semester Coverage: The book has 14 chapters and 60 sections (each with several dozen exercises) making it usable for either a one or two semester course. The section lengths are just right for coverage in one day. The book uses an example-abstract approach vs. an abstract axiom-example approach. This means, for example, that the definition of group is delayed a few weeks into the semester. Personally, I prefer such an approach and I think the students, exposed to an axiomatic approach for the first time, find it easier.
Non-standard Applications: Every author tries to include non-standard applications, that is, applications of modern algebra not found in almost all other text books. This book emphasizes a) the 15 puzzle, b) the RSA algorithm, c) Dedekind ideal theory, and (as already mentioned) the historically motivated d) solvability of equations and e) geometric constructibility.
I have never seen another modern algebra book with a presentation of the quadratic reciprocity law. True to the book’s spirit, both the historical (mathematical theorem with the second most proofs) and aesthetic (the golden theorem) aspects of quadratic reciprocity are mentioned. The law is presented with accompanying diagrams and computational exercises showing the theorem’s power. Of course, instructors who wish to can comfortably omit teaching the “Number Theory” chapter.
I found two topics lacking in the book.
Russell Jay Hendel, RHendel@Towson.Edu, holds a Ph.D. in theoretical mathematics and an Associateship from the Society of Actuaries. He teaches at Towson University. His interests include discrete number theory, applications of technology to education, problem writing, actuarial science and the interaction between mathematics, art and poetry.
Preface ix
1 The Early History 1
1.1 The Breakthrough 1
2 Complex Numbers 9
2.1 Rational Functions of Complex Numbers 9
2.2 Complex Roots 17
2.3 Solvability by Radicals I 23
2.4 Ruler and Compass Constructibility 26
2.5 Orders of Roots of Unity 36
2.6 The Existence of Complex Numbers* 38
3 Solutions of Equations 45
3.1 The Cubic Formula 45
3.2 Solvability by Radicals II 49
3.3 Other Types of Solutions* 50
4 Modular Arithmetic 57
4.1 Modular Addition, Subtraction, and Multiplication 57
4.2 The Euclidean Algorithm and Modular Inverses 62
4.3 Radicals in Modular Arithmetic* 69
4.4 The Fundamental Theorem of Arithmetic* 70
5 The Binomial Theorem and Modular Powers 75
5.1 The Binomial Theorem 75
5.2 Fermat's Theorem and Modular Exponents 85
5.3 The Multinomial Theorem* 90
5.4 The Euler φ-Function* 92
6 Polynomials Over a Field 99
6.1 Fields and Their Polynomials 99
6.2 The Factorization of Polynomials 107
6.3 The Euclidean Algorithm for Polynomials 113
6.4 Elementary Symmetric Polynomials* 119
6.5 Lagrange's Solution of the Quartic Equation* 125
7 Galois Fields 131
7.1 Galois's Construction of His Fields 131
7.2 The Galois Polynomial 139
7.3 The Primitive Element Theorem 144
7.4 On the Variety of Galois Fields* 147
8 Permutations 155
8.1 Permuting the Variables of a Function I 155
8.2 Permutations 158
8.3 Permuting the Variables of a Function II 166
8.4 The Parity of a Permutation 169
9 Groups 183
9.1 Permutation Groups 183
9.2 Abstract Groups 192
9.3 Isomorphisms of Groups and Orders of Elements 199
9.4 Subgroups and Their Orders 206
9.5 Cyclic Groups and Subgroups 215
9.6 Cayley's Theorem 218
10 Quotient Groups and their Uses 225
10.1 Quotient Groups 225
10.2 Group Homomorphisms 234
10.3 The Rigorous Construction of Fields 240
10.4 Galois Groups and Resolvability of Equations 253
11 Topics in Elementary Group Theory 261
11.1 The Direct Product of Groups 261
11.2 More Classifications 265
12 Number Theory 273
12.1 Pythagorean triples 273
12.2 Sums of two squares 278
12.3 Quadratic Reciprocity 285
12.4 The Gaussian Integers 293
12.5 Eulerian integers and others 304
12.6 What is the essence of primality? 310
13 The Arithmetic of Ideals 317
13.1 Preliminaries 317
13.2 Integers of a Quadratic Field 319
13.3 Ideals 322
13.4 Cancelation of Ideals 337
13.5 Norms of Ideals 341
13.6 Prime Ideals and Unique Factorization 343
13.7 Constructing Prime Ideals 347
14 Abstract Rings 355
14.1 Rings 355
14.2 Ideals 358
14.3 Domains 361
14.4 Quotients of Rings 367
A Excerpts: Al-Khwarizmi 377
B Excerpts: Cardano 383
C Excerpts: Abel 389
D Excerpts: Galois 395
E Excerpts: Cayley 401
F Mathematical Induction 405
G Logic, Predicates, Sets and Functions 413
G.1 Truth Tables 413
G.2 Modeling Implication 415
G.3 Predicates and their Negation 418
G.4 Two Applications 419
G.5 Sets 421
G.6 Functions 422
Biographies 427
Bibliography 431
Solutions to Selected Exercises 433
Index 440
Notation 444