On Évariste Galois

"An excessive desire for conciseness was the cause of this fault which one must try to avoid when writing on the mysterious abstractions of pure Algebra. Clarity is indeed an absolute necessity. [...] Galois too often neglected this precept." (Joseph Liouville, ["Avertissement" to "Oeuvres de Galois", 1846)

"It is perhaps less well known that [Galois] had also, without any possible doubt, discovered the essentials of the theory of abelian integrals, as Riemann would develop it 25 years later. By what route did he arrive at these conclusions? The fragments of calculations in Analysis found among his papers do not seem to permit much of an answer to that question, but there is room to imagine that he must have been very close to the idea of the Riemann surface associated with an algebraic function, and that such an idea must also be fundamental in his investigations into what he calls the 'théorie de l'ambiguïté'." (Jean Dieudonné, "Preface" to Ecrits et mémoires ďEvariste Galois

"Having found a method differing from that of Ferrari for reducing the solution of the general biquadratic equation to that of a cubic equation, Euler had the idea that he could reduce the problem of the quintic equation to that of solving a biquadratic, and Lagrange made the same attempt. The failures of such able mathematicians led to the belief that such a reduction might be impossible. The first noteworthy attempt to prove that an equation of the fifth degree could not be solved by algebraic methods is due to Ruffini (1803, 1805), although it had already been considered by Gauss. [...] The modern theory of equations is commonly said to date from Abel and Galois. [...] Abel showed that the roots of a general quintic equation cannot be expressed in terms of its coefficients by means of radicals." (David E Smith, "History of Mathematics", 1925)

"It is most remarkable that two men as different in character and outlook as Abel and Galois should have been interested in the same problem and should have attacked it by similar methods. Both approached the problem of the quintic equation in the conviction that a solution by radicals was possible; Abel at eighteen, Galois at sixteen. In fact, both thought for a while that they had discovered such a solution; both soon realized their error and attacked the problem by new methods." (Tobias Dantzig, "Number: The Language of Science", 1930)

"Since my mathematical youth, I have been under the spell of the classical theory of Galois. This charm has forced me to return to it again and again." (Mario Livio, "The Equation that Couldn't Be Solved: How Mathematical Genius Discovered the Language of Symmetry", 2005)

"Like moonlight itself, Monstrous Moonshine is an indirect phenomenon. Just as in the theory of moonlight one must introduce the sun, so in the theory of Moonshine one must go well beyond the Monster. Much as a book discussing moonlight may include paragraphs on sunsets or comet tails, so do we discuss fusion rings, Galois actions and knot invariants." (Terry Gannon, "Moonshine Beyond the Monster: The Bridge Connecting Algebra, Modular Forms and Physics", 2006)