Galois' theory of algebraic equations (Q2734150)

This book is based on a course which the author repeatedly taught at the Université Catholique de Louvain, Belgium, during the period from 1978 to 1989. The first published version of it appeared in 1988, under the same title, in the series ``Longman Scientific \& Technical'', Harlow, Wiley, New York (1988; Zbl 0663.12020). As the author points out, the present new edition does not differ substantially from the original Longman edition, although the wording of the text has been recast in a few places. NEWLINENEWLINENEWLINEIt is very gratifying to see that this very particular course on the theory of polynomial equations has been made available again, because it was and is indeed a methodological masterpiece within the vast existing literature on this subject. Namely, in spite of the title, the main subject of these lectures is neither abstract algebra nor its history, but a special part of mathematical methodology. The author's aim is to convey to the mathematically less experienced reader an idea of how mathematics is made, and what an amount of knowledge it can lead to. The theme used as an illustration for general methodology in mathematics is the theory of polynomial equations. For the purpose of these lectures, the theory of equations is an ideal topic in several respects. First, it is completely elementary, requiring virtually no mathematical background for the statement of its problems, and yet it leads to profound ideas and to fundamental concepts of modern algebra. Secondly, the theory of equations underwent a very long and eventful evolution, in the course of which the close interplay between both the individual ingenuity of several outstanding researchers and the collective experience of generations of mathematicians led to a complete understanding of the subject. NEWLINENEWLINENEWLINEThird, the development of the theory of algebraic equations is also very instructive, from the methodological point of view, in that it distinctly reflects the relationship between the general theory (Cardano, Tschirnhaus, Lagrange, Abel, Galois) and the attempts at significant examples (Vienta, de Moivre, Vandermonde, Gauss). NEWLINENEWLINENEWLINEAlong these lines, the author thoroughly reviews and discusses the main stages of the evolution of the theory of algebraic equations, from its origins in ancient times to its completion by Evariste Galois around 1830. As a consequence of emphasis on historical evolution, the exposition of mathematical facts in this book is genetic rather than systematic, which means that it primarily aims to retrace the concatenation of ideas by following (roughly) their chronological order of occurrence. Therefore, results that are logically close to each other may be scattered in different chapters, and some topics are discussed several times, in various contexts, instead of being given a unique definitive account. As the author points out, the expected reward for these circumlocutions is that the reader might gain a better insight into the inner workings of the theory, which prompted it to evolve the way it did. NEWLINENEWLINENEWLINEThe text is subdivided into fifteen chapters. NEWLINENEWLINENEWLINEChapter 1 describes the role of quadratic equations in Babylonian, Greek, and Arabic algebra. Chapter 2 discusses the attempts at solving cubic equations, ending up with Cardano's formula and the developments arising from it. Chapter 3 deals with quartic equations, in particular with Ferrari's method. Chapter 4 illustrates the rise of symbolic algebra and the creation of polynomials, whereas Chapter 5 gives the modern approach to polynomials over a ring (or field). Chapter 6 returns to cubic and quartic equations by describing the solution methods of Vieta, Descartes, Tschirnhaus, and others. Chapter 7 discusses roots of unity for cyclotomic polynomials, together with de Moivre's formula for complex numbers and the work of Leibniz and Newton on the summation of series. Chapter 8 gives an introduction to symmetric functions, including Waring's method, the concept of discriminant, and Euler's results on perfect squares. Chapter 9 is devoted to the ``Fundamental theorem of algebra'', containing Girard's proof of the existence of the splitting field of a polynomial and the proof of the fundamental theorem via splitting fields and symmetric polynomials. Chapter 10 explains Lagrange's systematizing contribution towards a general theory of polynomial equations and provides, at the end, first results of group theory and Galois theory. NEWLINENEWLINENEWLINEChapter 11 honors the work of A.-T. Vandermonde within these developments, whilst Chapter 12 describes, in great detail, C. F. Gauss's ingenious results on cyclotomic equations, the problem of their solvability by radicals, and the construction of regular polygons by ruler and compass. Chapter 13 continues the discussion of this development by switching over to the works of Ruffini and Abel. Radical extensions, Abel's theorem on natural irrationalities, and the proof of the unsolvability of general equations of degree higher than four by radicals are the main topics of this chapter. Chapter 14 then culminates in comprehensively describing E. Galois's completion of the theory of algebraic equations in one variable. This chapter, being the longest and most abstract one in the entire book, really does general Galois theory of field extensions, with all the algebraic tools needed for deriving the results on the problem of solvability of polynomial equations by radicals. NEWLINENEWLINENEWLINEThe final Chapter 15 is entitled ``Epilogue'' and provides, as an appendix to the foregoing text on algebraic equations a proof of the ``Fundamental theorem of Galois theory''. This leads from the theory of polynomial equations of one variable over to the general Galois theory of fields, the latter of which being beyond the scope (and intention) of this methodologically oriented textbook. NEWLINENEWLINENEWLINEAltogether, this work is a very welcome addition to the ample literature on classical Galois theory, especially so from the viewpoints of culture, history, and methodology in mathematical science. The author has done a great service to the entire mathematical community.