Algebra from al Khwarizmi to Galois, 830--1830 (Q779708)

Let me start by providing the reader with a list of books (not mentioned by the author in the paper under review) whose contents contain serious studies in the history of algebra, namely \begin{itemize} \item \textit{Peter M. Neumann}, The mathematical writings of Évariste Galois [Zbl 1237.01011]; \item \textit{Dirk J. Struik}, A concise history of mathematics [Zbl 0032.09701]; \item \textit{Dirk J. Struik} (ed.), A source book in mathematics (1200--1800) [Zbl 0205.29202]; \item \textit{Bartel van der Waerden}, A history of algebra. From al-Khwārizmī to Emmy Noether [Zbl 0569.01001]; \item \textit{Bartel van der Waerden}, Geometry and algebra in ancient civilizations [Zbl 0534.01001]; \item \textit{Emil Artin}, Galoissche Theorie [Zbl 0086.25703]; \item \textit{Ian Stewart}, Galois theory [Zbl 0269.12104]. \end{itemize} Very important for the history of mathematics, historians practicing in mathematics, and personalities in the mathematical community is the \begin{itemize} \item MacTutor History of Mathematics Archive (\url{https://mathshistory.st-andrews.ac.uk/}). \end{itemize} Now, let us deal with the paper under review. The aim of the author is to sketch a condensed development of solving polynomial equations by means of radicals (in our terminology). On Page~614 he speaks about al-Khwarizmi and his efforts in dealing with cubic equations. On Page~615 the author connects cubic equations with geometrical problems such as the problem of trisecting an angle by particular ``apparatus'' or ``means'', due to the Greeks and the Islamic authors Omar Khayyam and al-Khwarizmi. And then, beginning with Page~617, he starts considering the works and attempts by the 16th and 17th-century mathematicians from Italy, see below. The paper ends with the story around Galois, Abel, Ruffini and the author gives some worked-out examples of what Galois theory should be, as well as there should exist a notion of the so-called Galois groups. Despite the author's enthusiasm about the subject, the reviewer feels it as his task to bring up some critical points, to wit: \begin{itemize} \item On Page~614, the author says (quoted): ``Yet [Galois, the rev.] answered a central question in algebra posed 1000 years before: For which equations [of \(n\)th degree, the rev.] is it possible to compute the unknown \(x\) by basic arithmetic operations and extraction of roots (of any order), and for which equations is this impossible? [\dots] it was the year 1832!'' Well, the Babylonians already did study about 2000 years before al-Khwarizmi (i.e., 3500 years ago) quadratic expressions of equations by means of geometry. \item On Page~615, the author obtains \(x=7\) as the positive solution of the equation \(x^2+21=10x\), stating that the other solution is negative thus not mentioned by al-Khwarizmi. However, \(x=3\) is a second positive solution. \item On Page~616, it is mentioned that Omar Khayyam and other Islamic mathematicians solved cubic equations by means of geometry, i.e., by intersecting conic sections. The author says (quoted): ``Concerning cubic equations in general, Omar Khayyam admits [\dots] that he [\dots] has not been able to find an arithmetic solution in the way al Khwārizmī did this for quadratic equations: `Perhaps someone else who comes after us may find it out in the case, when there are not only the first three classes of known powers, namely the number, the unknown and its square.{'}'' Then, the author says: ``It took more than 400 years to happen!'' I checked in several sources what is going on here effectively. I found that Khayyam in his sentence ``Perhaps \dots{} square'' meant he was not able to provide a solution of the general cubic equation, let alone in finding cubic equations admitting three \textit{real} roots. Subsequently, the author goes on with del Ferro, Cardano, Tartaglia, Ferrari, regarding cubic and biquadratic equations, negative numbers and complex numbers (Bombelli). \item On Pages~620 and 621, the author tries to inform the reader in a compressed way what Galois did in his research, by introducing notions such as ``solvability'', ``Galois groups'' and the like. But especially here see any of the books in the list I mentioned, and of course the MacTutor Archive. In the footnote on Page~621, the sentences on the structure of the Galois group of the equation \(x^n-a=0\) are confusing, and in error: the order of the Galois group in general is not equal to \(n\), the group is not always cyclic. Indeed, the Galois group of \(x^4-2\in\mathbb{Q}[x]\) is dihedral of order 8, that of \(x^3-2\in\mathbb{Q}[x]\) is isomorphic to the symmetric group of three symbols, of order~6. \end{itemize} In conclusion, the paper under review is not flawless; it is hardly possible to compress the developments of the subject in the last 1500 years or so within twelve pages.