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Galois' Theory of Algebraic Equations

Jean-Pierre Tignol
World Scientific
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The Basic Library List Committee recommends this book for acquisition by undergraduate mathematics libraries.

[Reviewed by
Mark Hunacek
, on

Back in the day, it was common for undergraduate mathematics departments to offer courses with a title like “The Theory of Equations”. To accommodate these courses, books with similar titles, with authors like Uspensky, Dickson, Borofsky, Cajori and MacDuffee, were easily found.

That was then, however, and this is now, and such courses have for the most part become extinct, supplanted by courses with titles like “Abstract Algebra II”. In one respect this is unfortunate, because the old books on the theory of equations covered a lot of material that is often never discussed at all in the more modern courses, such as the formulas for the roots of cubic and quartic polynomials. This material is not only mathematically interesting, but has obvious historical interest as well, because the development of the theory of equations over the years is, to some considerable extent, the history of algebra. (These topics may be covered in a history of mathematics course, but, if the experience at my university is at all typical, these courses aren’t offered as much as they used to be, either.) The result of all this is that students nowadays graduate as mathematics majors with a lot of modern definitions under their belt, but these definitions are taught from a modern perspective; hence, while many of them can tell you what a “group” is, they really have no idea how this definition came to be. Many of them have never heard of people like Cardano, or how his work on cubic equations fits into the development of algebra.

Books like the one under review help to remedy this situation. Tignol’s book does not fit neatly into the current undergraduate mathematics curriculum, but it is one that any person with an interest in algebra or history will want to read, or, even better, own. Despite its title, it does far more than just introduce Galois theory, but instead serves as a broad survey of how mathematical ideas helped shape algebra over the years. It is written so as to be accessible to undergraduates, and is a real accomplishment.

The book traces the history of the theory of equations from ancient times to the work of Galois, following a chronological development. An initial short chapter looks at how the ancient Babylonians, Greeks and Arabs handled equations. From here we move forward to the 1500s and the time of Cardano, Tartaglia, and Ferrari, who developed formulas for the solution of cubic and quartic polynomial equations. The discussion is both historical and mathematical; although Tignol discusses, for example, the feud between Cardano and Tartaglia, he also provides details of the methods involved and explains, for example, how Cardano’s formula can lead, even in the case of real roots, to expressions involving the square root of negative numbers. Historically, this is the impetus for the discovery of complex numbers, and the author discusses Bombelli’s contributions in this area.

Cardano did not have at his disposal the modern notation that we use today, and his book on equations, Ars Magna, is largely written in prose. Chapters 4 and 5 of the book under review discuss polynomials and the rise of modern notation. A major contributor here was Viète, whose work in this area is discussed; Viète also was one of several people who worked on alternative approaches to cubic and quartic equations, and these efforts are described in chapter 6.

Chapter 7 is the first of several chapters that involve cyclotomic polynomials. Beginning with the historical evolution of DeMoivre’s formula, it proceeds through roots of unity and then, in later chapters, Gauss’s work on roots of unity and cyclotomic polynomials. It is shown, for example, that cyclotomic polynomials are solvable by radicals. The historical evolution of these ideas involves the Fundamental Theorem of Algebra, which is also discussed and proved (chapter 9).

Of course, it is well-known among mathematicians that, unlike cyclotomic polynomials, the general polynomial of degree 5 or greater is not solvable by radicals. This result is credited to Ruffini and Abel (whose work is discussed in chapter 13); however, the author is careful to point out that this result builds on previous work of Lagrange, which is described in chapter 10.

As a result of all this, we are in a situation where, to quote the author:

two major classes of equations have been treated thoroughly, with divergent results: the cyclotomic equations are solvable by radicals in any degree, while general equations of degree at least five are not. Thus, the obvious question arises: which equations are solvable by radicals?

And, of course, as the author points out, after discussing preliminary work of Abel that might have resulted in a complete answer had Abel not died prematurely, the “honor of finding a complete solution to the problem eventually fell to another young genius, Évariste Galois”. Chapter 14 of the text is a summary of Galois’s work, using (as the author has not hesitated to do throughout the text) modern notation and language, but keeping quite true to the way in which Galois dealt with these issues.

Chapter 14, in fact, constitutes the most significant change between the first and second editions of this book. The author describes in the Preface how, in doing work in an unrelated project, he noticed a strong analogy with certain constructions used by Galois. As a result, he says, the chapter on Galois “has been completely rewritten to take advantage of this viewpoint. The exposition is now much closer to his memoir, and remarkably elementary.” He is correct in that very little in the way of technical mathematical knowledge is required for this chapter, but, as he himself candidly acknowledges, “‘elementary’ is not the same as ‘easy’ … Galois’s elementary arguments are remarkably ingenious and sometimes quite intricate.” To help pave the way for his explication of these arguments, Tignol begins with a very clear and informative summary and overview of what is going on, a discussion which itself struck me as very valuable.

Chapter 14 is the last “official” chapter of the book, but it is followed by an Epilogue, in which the author does several things. First, he surveys some post-Galois history of abstract algebra, including a nice exposition of Emil Artin’s approach to the subject and giving Artin’s statement and proof of the Fundamental Theorem of Galois Theory. (If you like getting things “from the horse’s mouth”, see the Dover edition Galois Theory: Lectures Delivered at the University of Notre Dame, reviewed in this column several months back.) Secondly, in a fairly sophisticated section of the Epilogue, Tignol elaborates on the comments that he made in the Preface and discusses the connection between Galois’s ideas and the Grothendieck theory of étale algebras. Though the author attempts to make this section self-contained, it is nonetheless something that a beginner would likely find daunting.

There is, however, a great deal of material in this book that should be comprehensible to a decently prepared undergraduate. This is also material that does not appear in very many other recent texts. Cooke’s Classical Algebra does have, particularly in its last hundred pages or so, substantial overlap with Tignol’s book, but Tignol devotes about three times as many pages to the subject and covers things that Cooke does not, such as the theory of roots of unity and, of course, the exposition of Galois’ memoir. (The omission of the former topic was noted in our earlier review of Cooke.)

So, to summarize, let me finish this review as I started it: anybody with an interest in algebra or the history of mathematics should look at this book. And of course it goes without saying that it belongs in any good university library.

Mark Hunacek ( teaches mathematics at Iowa State University.

  • Quadratic Equations
  • Cubic Equations
  • Quartic Equations
  • The Creation of Polynomials
  • A Modern Approach to Polynomials
  • Alternative Methods for Cubic and Quartic Equations
  • Roots of Unity
  • Symmetric Functions
  • The Fundamental Theorem of Algebra
  • Lagrange
  • Vandermonde
  • Gauss on Cyclotomic Equations
  • Ruffini and Abel on General Equations
  • Galois
  • Epilogue