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Classical Algebra: Its Nature, Origins, and Uses

Roger Cooke
John Wiley
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The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
Fernando Q. Gouvêa
, on

Two books were in Roger Cooke’s mind, and they struggled together within him. That may be the best explanation for the peculiar shape of this very interesting book. Someone influenced by Biblical “higher criticism” might well want to postulate two underlying sources. Let’s call them A (for “algebra”) and E (for “equations”).

Source A seems to have been a general introduction to “classical algebra,” by which I mean “the algebra one studies in school,” or, perhaps, “the algebra that existed before Galois.” The parts of the book that most display the influence of A are the first four chapters, which focus on what algebra is and what it is for. The discussion uses historical examples, but it is not really a historical account. Rather, it tries to answer the questions one can imagine intelligent but non-mathematical college students asking when told that they must take a “college algebra” course.

Source E, on the other hand, was about the classical theory of polynomial equations in one variable, a subject that has pretty much disappeared from the curriculum. Cooke’s account is historical, though, as he himself points out, it should be considered an investigation of “heritage” rather than pure history. The interest is in seeing how we got to where we are now, following the thread from quadratic equations to the unsolvability of the quintic, Galois theory, and the first glimpses of “modern Algebra.”

(Cooke opts for the “classical” versus “modern” nomenclature instead of using “abstract algebra.” This has the advantage of providing fairly neutral names for both areas, since there is no natural counterpart to “abstract” — certainly not “concrete”! Of course, if we stick to “modern” to describe events circa 1860–1940, we’ll have to find another name for more recent stuff. Perhaps “postmodern”?)

Both A and E could have been developed on their own into very interesting books. Blending them creates both a practical problem and some more substantive issues.

The practical problem is the obvious one: who is the intended audience? Mathematicians trained in abstract algebra and Galois theory will be able to follow the whole book, but might be less interested in the first few chapters. Very few students, and perhaps also few teachers, will be able to handle the (“elementary” but difficult) theory of resultants discussed in the later chapters, for example. This is a pity, since the discussion in the first four chapters has the potential to reach a much broader audience.

The more substantive problem is that the blending of sources has allowed E to influence A in a way that weakens the argument. Because E focuses exclusively on polynomial equations in one variable, the argument in A also tries to stick to that part of the subject. That seems like a mistake to me. For one thing, there is so much more out there, and the possibilities of linear algebra and polynomials in many variables are too rich to ignore, particularly when discussing whether algebra is useful.

The “why algebra” question is a good one. Many historians (Cooke himself included) have remarked on the artificiality of the problems that appear in early texts on algebra, and it is no secret that this artificiality persists today. As Cooke points out, how can one know that “A father is three times as old as his son today, and in ten years he will be twice his son’s age” without actually knowing their ages first? Such problems seem to make algebra simply the source of innumerable pointless riddles.

Cooke’s answer is that algebra allows us to see the further truths hidden in known arithmetical relationships. By setting up symbolic representations of such relationships and teaching us how to combine, manipulate, and compare them, algebra unearths truths that are hidden in things we already know or assume. This is certainly both interesting and true, but it is most true precisely when we are talking about relationships that involve multiple variables. In fact, Cooke’s examples, which mostly derive from physics, all involve multiple variables and relationships between them.

This hardly answers, however, the more specific question about the usefulness of equations in one variable. I expected to see examples of practical problems that lead to such equations, but they never showed up. One might speculate that we are missing parts of source A in which further aspects of algebra and its utility are discussed. Instead, source E becomes dominant at about this point.

There is less to criticize in  source E. It is an excellent account of how the theory of equations developed from the early work of Italian algebraists to the work of Abel and Galois. At times, I wish Cooke had gone into more detail (did source E get shortened to make space for the material from A?), but this is rich — and alas, nowadays not well known — material.

The one suggestion I would make for a new edition would be to include a discussion of Gauss’s work on roots of unity. This would help answer an annoying questions that is rarely treated in texts on Galois theory: when treating solution by radicals, one typically just includes roots of unity in the game, on the grounds that “the n-th root of 1” is a radical expression. But one could ask for more, i.e., one could ask for a proof that the primitive n-th root of 1 can be expressed as x + iy, where x and y are expressible by radicals of non-units. This is easy to see for n = 3 and n = 5, where we can find such a representation “by hand.” Gauss’s work is the key to giving a proof.

Despite these concerns, this is a book that can contribute a lot. Anyone teaching classical algebra, whether in high school or in college, will find a lot to think about here. They might find some of the later chapters tough sledding, and should skim as necessary. For those of us who teach modern algebra and Galois theory, this book will play a different, but still fundamental role: it will help us understand, and explain to our students, how we got from equations in one variable to groups and fields. Classical Algebra deserves a large audience, and should be in every undergraduate library.

Fernando Q. Gouvêa is Carter Professor of Mathematics at Colby College, where he teaches both history of mathematics and “abstract” algebra.

Preface ix

 Part 1. Numbers and Equations

 Lesson 1. What Algebra Is

1. Numbers in disguise

1.1. Classical and modern algebra

2. Arithmetic and algebra

3. The environment of algebra: Number systems

4. Important concepts and principles in this lesson

5. Problems and questions

6. Further reading


Lesson 2. Equations and Their Solutions

1. Polynomial equations, coefficients, and roots

1.1. Geometric interpretations

2. The classification of equations

2.1. Diophantine equations

3. Numerical and formulaic approaches to equations

3.1. The numerical approach

3.2. The formulaic approach

4. Important concepts and principles in this lesson

5. Problems and questions

6. Further reading


Lesson 3. Where Algebra Comes From

1. An Egyptian problem

2. A Mesopotamian problem

3. A Chinese problem

4. An Arabic problem

5. A Japanese problem

6. Problems and questions

7. Further reading


Lesson 4. Why Algebra Is Important

1. Example: An ideal pendulum

2. Problems and questions

3. Further reading


Lesson 5. Numerical Solution of Equations

1. A simple but crude method

2. Ancient Chinese methods of calculating

2.1. A linear problem in three unknowns

3. Systems of linear equations

4. Polynomial equations

4.1. Noninteger solutions

5. The cubic equation

6. Problems and questions

7. Further reading


Part 2. The Formulaic Approach to Equations

 Lesson 6. Combinatoric Solutions I: Quadratic Equations

1. Why not set up tables of solutions?

2. The quadratic formula

3. Problems and questions

4. Further reading


Lesson 7. Combinatoric Solutions II: Cubic Equations

1. Reduction from four parameters to one

2. Graphical solutions of cubic equations

3. Efforts to find a cubic formula

3.1. Cube roots of complex numbers

4. Alternative forms of the cubic formula

5. The irreducible case

5.1. Imaginary numbers

6. Problems and questions

7. Further reading


Part 3. Resolvents

 Lesson 8. From Combinatorics to Resolvents

1. Solution of the irreducible case using complex numbers

2. The quartic equation

3. Viµete's solution of the irreducible case of the cubic

3.1. Comparison of the Viµete and Cardano solutions

4. The Tschirnhaus solution of the cubic equation

5. Lagrange's reflections on the cubic equation

5.1. The cubic formula in terms of the roots

5.2. A test case: The quartic

6. Problems and questions

7. Further reading


Lesson 9. The Search for Resolvents

1. Coefficients and roots

2. A unified approach to equations of all degrees

2.1. A resolvent for the cubic equation

3. A resolvent for the general quartic equation

4. The state of polynomial algebra in 1770

4.1. Seeking a resolvent for the quintic

5. Permutations enter algebra

6. Permutations of the variables in a function

6.1. Two-valued functions

7. Problems and questions

8. Further reading


Part 4. Abstract Algebra

Lesson 10. Existence and Constructibility of Roots

1. Proof that the complex numbers are algebraically closed

2. Solution by radicals: General considerations

2.1. The quadratic formula

2.2. The cubic formula

2.3. Algebraic functions and algebraic formulas

3. Abel's proof

3.1. Taking the formula apart

3.2. The last step in the proof

3.3. The verdict on Abel's proof

4. Problems and questions

5. Further reading


Lesson 11. The Breakthrough: Galois Theory

1. An example of a solving an equation by radicals

2. Field automorphisms and permutations of roots

2.1. Subgroups and cosets

2.2. Normal subgroups and quotient groups

2.3. Further analysis of the cubic equation

2.4. Why the cubic formula must have the form it does

2.5. Why the roots of unity are important

2.6. The birth of Galois theory

3. A sketch of Galois theory

4. Solution by radicals

4.1. Abel's theorem

5. Some simple examples for practice

6. The story of polynomial algebra: a recap

7. Problems and questions

8. Further reading

Epilogue: Modern Algebra

1. Groups

2. Rings

2.1. Associative rings

2.2. Lie rings

2.3. Special classes of rings

3. Division rings and fields

4. Vector spaces and related structures

4.1. Modules

4.2. Algebras

5. Conclusion


Appendix: Some Facts about Polynomials

Answers to the Problems and Questions

Subject Index

Name Index