Abel's Proof: An Essay on the Sources and Meaning of Mathematical Unsolvability

This fascinating book owes its existence to the author's desire for a "basic insight" into the general unsolvability of polynomial equations of degree five or higher. Peter Pesic is Tutor and Musician-in-Residence at St. John's College in Santa Fe (a college that prefers not to use professorial labels). His Ph. D. is in physics, but his biography on the St. John's website suggests that he is continuing a lifelong project as Polymath-in-Training. His other two books for MIT Press concern physics, philosophy, and literature in one instance, and the "hidden meaning of science" in the other. He has edited works of Maxwell and Planck; lists 32 papers on scientific, mathematical, and musical topics; as pianist has performed cycles of the complete sonatas or complete keyboard music of Bach, Beethoven, Haydn, Mozart, Schubert, and the Second Viennese School (Schoenberg, Berg, and Webern); and so on.

A brief Introduction reviews some terminology (degree; quadratic, cubic, ...) and reveals the great divide between degrees 4 and 5, then states a question that the book intends to answer: "How can a search for solutions yield the unsolvable?" Chapter 1 (The Scandal of the Irrational) traces companion notions of irrationality and incommensurability to their Greek origins: Pythagorean discoveries, Plato's use of mathematical ideas in some of the Dialogues, and Euclid's Book X investigation of the concept of irrationality.

Chapter 2 (Controversy and Coefficients) moves quickly through Babylonian and then Arabic developments (both seen as algebra without the symbols), Fibonacci, Pacioli, and the cubic and quartic triumphs of the sixteenth-century Italians, to the notational innovations of Viète. Chapter 3 (Impossibilities and Imaginaries) notes Viète's conviction that all polynomial equations should be solvable by the same methods that worked for those of degrees 2, 3, and 4, a view disputed by Kepler but shared by Descartes, whose work gave complex numbers a kind of legitimacy.

Chapter 4 (Spirals and Seashores) describes Newton's identities relating roots and coefficients (generalizing earlier identities due to Girard), and a lemma from the Principia demonstrating the impossibility of expressing the area of an "oval" by a finite algebraic formula, which Pesic sees as an early warning of the possible unsolvability of the quintic. On the other hand, work of Tschirnhaus and later E. S. Bring shows that being able to solve a quintic that is missing terms of degrees 2, 3, and 4 is sufficient to solve the general quintic. The chapter ends with one of Gauss' proofs of the Fundamental Theorem of Algebra.

Chapter 5 (Premonitions and Permutations), the last chapter before an outline of Abel's proof, discusses two important landmarks. One is Lagrange's 1771 paper (somewhat anticipated by Vandermonde) on the number of different values that can be taken on by the resolvent of an equation when the roots are permuted in all possible ways; this number is less than n for equations of degree n = 3 and 4, but greater than n for degree 5. Lagrange interprets this as meaning that a solution of the quintic (and presumably beyond) will require methods different from those working for cubics and quartics, although he remained optimistic about an ultimate solution of the quintic. But from 1799 to 1813 the physician and mathematician Ruffini presented six versions of a proof of the quintic's unsolvability, a proof that was difficult for most mathematiicans of the time to understand (although Cauchy expressed his admiration) and was ultimately seen to be incomplete.

Chapter 6 (Abel's Proof) presents a few biographical details about Abel, including an attempted quintic solution, and then outlines Abel's 1824 proof of the unsolvability of the quintic. Pesic comments that this proof "...in many ways is close to Ruffini's proof, although it fills in an important gap that Ruffini had not noticed." Assuming the quintic is solvable, Abel deduces the form of any solution, shows that it must involve rational functions of the roots (this is Ruffini's missing step), then uses a theorem of Cauchy, on the values a rational function of five quantities can assume when those quantities are permuted, to reach a contradiction.

The remaining four chapters are anticlimactic in a way, but nevertheless well worth reading. They exist due to Pesic's feeling that Abel's argument "...still seems opaque" and his need to "...seek the heart of Abel's proof." Chapter 7 (Abel and Galois) recounts Abel's travels, including a satisfactory publishing relationship with Crelle (as opposed to chilly receptions by Cauchy and Legendre). Abel's "...insight [connecting] solvability with commutativity" is followed (after biographical details of Galois' tragic life) by Galois' reformulation of the mathematical situation in what we now recognize as group theory. Chapter 8 (Seeing Symmetries) uses a clever model of dancers changing places to develop fundamental facts about the first few symmetric groups and their normal subgroups. The 2-dancer model is motivated by the quadratic formula, and the author concludes that "Here emerges our basic insight: Solving an equation corresponds to a certain commutative symmetry." The 3-dancer and 4-dancer models also demonstrate a pattern (in the language that concludes the chapter) of nested normal subgroups linked in an abelian way, signalling that the cubic and quartic equations can be solved. Needless to say, the pattern breaks down for 5 dancers.

Chapter 9 (The Order of Things) consists of general comments on various developments in algebra, with special attention to noncommutativity (in vector spaces, matrix algebra, and quantum theory, for example) and to the importance of the group notion, while Chapter 10 (Solving the Unsolvable) offers the author's philosophical reflections on the meaning of the crises involving the infinite, from the Greek discovery of irrationals to Abel's discovery of unsolvability.

Three appendices flesh out the outline in Chapter 6, and include expansions of what Pesic calls "terse assertions" of the 1824 paper in Abel's 1826 version. Appendix A is a translation of the 1824 paper, with valuable running commentary supplied by Pesic. Appendix B is Abel's elaboration on the form that a quintic solution must take, and Appendix C describes a theorem of Cauchy on permutations that is crucial in the overall proof. These are followed by 20-some pages of Notes, many giving references for statements made in the text or suggestions for supplementary reading, but there are also a few explanations of material that the author chose not to include in the text, such as the story about normal subgroups, quotient groups, and solvable groups.

This book is a wonderfully accessible account of nonsolvability, for those with appropriate background. While the author has chosen to isolate certain arguments and examples in boxes that (he says in the Introduction) "can be skipped without guilt," one wonders if a reader for whom "the quadratic formula is a dim memory, at best" (Introduction again) would be able to complete this journey. There are some strange choices of boxed and unboxed items. Babylonian and Arabic solutions of quadratics (in modern dress) are confined to boxes in Chapter 2, while in Chapter 9, Hamilton's quaternions and ideas of Planck and Dirac on quantum theory are discussed right out in public. But then, one person's box-to-be-skipped can be another's need-to-know material. Overall the book is a splendid achievement.

David Graves (dgraves@elmira.edu) is Professor of Mathematics at Elmira College, where he is active as a pianist, and has taught courses in cryptology, opera, and history of astronomy as well as the usual run of mathematics courses.