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Introductory Modern Algebra: A Historical Approach

Saul Stahl
Publication Date: 
Number of Pages: 
[Reviewed by
Russell Jay Hendel
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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.

  • 10 of them are of the form “compute the group of symmetries of x1/x2 + x3/x4
  • 12 of them are of the form “compute the product and find the axes and angles of the rotations of the following two rotations of the tetrahedron in Figure 9.3: A = (1 2 3) and B = (2 4 3)”
  • 3 of them ask about the dihedral group Dn (e.g. how many elements does the group have; how many have order 2, etc.)
  • 4 of them are of the form “Describe the vertex symmetries of the cube in Figure 9.5”
  • 6 of them are “verification proofs”, for example, “show that all even permutations form a group”
  • 2 of them are more serious proofs, for example, “Prove that for every positive integer k there is a polygon whose group of vertex symmetries contains k elements.”

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.

  • The book has a standard appendix on mathematical induction. There are 18 exercises; a) there are routine exercises such as proof of the sum of squares or cubes of the first n integers; b) there are also inequality proofs such as “prove (by induction!) that 2n > n2 for n > 4” ; c) there are number theory exercises such as “prove (by induction!) that 11n+2+122n+1 is divisible by 133” ; and there are d) geometric and e) integral exercises.
  • The exercises on permutations deal with a) traditional permutations of numbers, b) permutations of multi-variable polynomials, c) permutations of geometric objects.
  • The exercises on polynomials are enriched by many exercises requesting factorizations over finite fields.

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.

  • The Beautiful Cayley Counting Theorem: At this I was surprised, since the Cayley theory lends itself to many clever and combinatorial computational exercises that would be consistent with the book’s goals. This is not a serious omission, however, unless the student target population consists of Chemistry majors. I would recommend to the author to include an additional chapter in edition 3 or 4.
  • The Sylow Theorems: I and many colleagues prefer to omit this topic anyway.

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