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Évariste Galois

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Évariste Galois (circa 1826)

Évariste Galois (October 25, 1811 – May 31, 1832) was a French mathematician, who, while still in his teens, developed the well-known Galois theory. Galois theory is capable to determine whether a polynomial with rational coefficient can be solved by radicals and give a clear insight about what kind of length ratio can be constructed by compass and straightedge, thereby solving the long-standing problems of solving a polynomial by radicals. His life is considered to be one of the most romantic in all of mathematics because of the contributions he has made in such a short span of life.

Quotes

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  • How to console oneself for having exhausted in one month the greatest source of happiness which is in man — of having exhausted it without happiness, without hope, being certain that one has drained it for life? Oh! come and preach peace after that! Come and ask men who suffer to take pity upon what is! Pity, never! Hatred, that is all. He who does not feel it deeply, this hatred of the present, cannot really have in him the love of the future. ...I like to doubt your cruel prophecy when you say that I shall not work any more. But I admit that it is not without likelihood. To be a savant, I should need to be that alone. My heart has revolted against my head. I do not add as you. do: It is a pity.
  • ... un auteur ne nuit jamais tant à ses lecteurs que quand il dissimule une difficulté.
    • ... an author never does more damage to his readers than when he hides a difficulty.
    • in the preface of Deux mémoires d'Analyse pure, October 8, 1831, edited by Jules Tannery (1908). Manuscrits de Évariste Galois. Gauthier-Villars. p. 27. 
  • Ne pleure pas, Alfred ! J'ai besoin de tout mon courage pour mourir à vingt ans !
    • Don't cry, Alfred! I need all my courage to die at twenty.
    • Quoted in: Léopold Infeld (1978) Whom the gods love: the story of Évariste Galois. p. 299.
  • Dès le commencement de ce siècle, l'algorithme avait atteint un degré de complication tel que tout progrès était devenu impossible par ce moyen, sans l'élégance que les géomètres modernes ont su imprimer à leurs recherches et au moyen de laquelle l'esprit saisit promptement et d'un seul coup un grand nombre d'opérations. Il est évident que l'élégance si vantée et à si juste titre n'a pas d'autre but. Du fait bien constaté que les efforts des géomètres les plus avancés ont pour objet l'élégance on peut donc conclure avec certitude qu'il devient de plus en plus nécessaire d'embrasser plusieurs opérations à la fois, parce que l'esprit n'a plus le temps de s'arrêter aux détails. ... Sauter à pieds joints sur les calculs, grouper les opérations, les classer suivant leurs difficultés et non suivant leurs formes; telle est, suivant moi, la mission des géomètres futurs; telle est la voie où je suis entré dans cet ouvrage.
    • Since the beginning of the century, computational procedures have become so complicated that any progress by those means has become impossible, without the elegance which modern mathematicians have brought to bear on their research, and by means of which the spirit comprehends quickly and in one step a great many computations. It is clear that elegance, so vaunted and so aptly named, can have no other purpose. From the well-established fact that the efforts of the most advanced geometers have elegance as their goal, we can therefore conclude with certainty that it is becoming increasingly necessary to encompass several operations at once, because the mind no longer has time to dwell on details. ... Go to the roots of these calculations! Group the operations. Classify them according to their complexities rather than their appearances! This, I believe, is the mission of future mathematicians. This is the road on which I am embarking in this work.
    • French version first published in Taton, René (1947) . "Les relations d'Évariste Galois avec les mathématiciens de son temps". Revue d'histoire des sciences et de leurs applications 1 (2): 114–130. ISSN 0048-7996.
    • English translation quoted in: Kiernan, B. Melvin (1971) . "The Development of Galois Theory from Lagrange to Artin". Archive for History of Exact Sciences 8 (1/2): 40–154. ISSN 0003-9519.
  • [This] science is the work of the human mind, which is destined rather to study than to know, to seek the truth rather than to find it.
  • Mais je n’ai pas le temps et mes idées ne sont pas encore bien développées sur ce terrain qui est immense. [But I don't have the time and my ideas are not yet well developed on this immense terrain.]
    • The Testamentary Letter of 29 May 1832
  • Il parait après cela qu'il n'y a aucun fruit à tirer de la solution que nous proposons. [It seems there is no fruit to be drawn from the solution we offer.]
    • Page 226, VI.4 Dossier 9: Preliminary discussion, folio 59a.
    • Galois was saying that his theory proved that there is no "fruit" of a general solution to the degree-five polynomial.

The mathematical writings of Évariste Galois (2011)

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edited by Peter M. Neumann.
  • Poisson, reading Galois' First Memoir, found the proof of Lemma III insufficient, and wrote in pencil the following comment.
  • La démonstration de ce lemme n’est pas suffisante; mais il est vrai d’après le No . 100 du mémoire de Lagrange, Berlin, 1771. [The proof of this lemma is insufficient; but it is true according to No . 100 of the memoir by Lagrange, Berlin, 1771.]
  • It angered Galois sufficiently that he wrote directly below it:
  • Nous avons transcrit textuellement la démonstration que nous avons donnée de ce lemme dans un mémoire présenté en 1830. Nous y joignons 154 IV The First Memoir comme document historique le note suivante qu’a cru ^devoir^ y apposer M. Poisson. On jugera. Note de l’auteur [We have faithfully transcribed the proof of this lemma that we have given in a memoir presented in 1830. We append as a historical document the following note which Mr Poisson believed he should add. Posterity will judge. Note by the author]

Quotes about Galois

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  • [In the first case, the problem is solving equations, including solving an algebraic equation with an unknown variable. In the second case, the problem is integrating differential equations.
    Galois was the first to recognize with absolute clarity how extraordinarily important the concepts of substitution group and invariant of a discontinuous group are for dealing with problems of the first kind.]
    Im ersten Falle hat man das Problem der Auflösung der Gleichungen, unter Anderm der Auflösung einer algebraischen Gleichung mit einer Unbekanuten. Im zweiten Falle hat man das Problem der Integration von Differentialgleichungen.
    Galois war der erste, der vollkommen klar erkannte, wie ausserordentlich wichtig die Begriffe Substiutionengruppe und Invariante einer discontinuirlichen Gruppe für die Behandlung von Problemen jener ersten Art sind.
    • Sophus Lie, Theorie der Transformationsgruppen (1893) Vorrede, p. XX. [Tr. Google.]
  • [We will not dwell further on the theory of transformation groups of a linear equation. We believe we have sufficiently demonstrated... the value of this theory, which is simply the natural extension to a question of analysis of the fruitful ideas introduced into algebra by Galois]
    Nous n'insisterons pas davantage sur la théorie des groupes de transformations d'une équation linéaire. Nous pensons avoir suffisamment montré... l'intérèt de cette théorie, qui n'est que l'extension bien naturelle à une question d'Analyse des idées si fécondes introduites en Algèbre par Galois.
  • Galois's introduction of imaginary roots of congruences has not only led to an important extension of the theory of numbers, but has given rise to wide generalizations of theorems which had been obtained in subjects like linear congruence groups by applying the ordinary theory of numbers.
  • It is... not true... that he proved and used the simplicity of alternating groups. He did not need to: he was much cleverer than that; his treatment of solubility of equations is... simpler and more elegant than... textbook tradition.

Evariste Galois (Oct 10, 1921)

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by George Sarton, The Scientific Monthly Vol. 13, pp. 363-375.
  • He was still a mere boy, yet within these short years he had accomplished enough to prove indubitably that he was one of the greatest mathematicians of all times.
  • [T]he soul of Galois will burn on throughout the ages and be a perpetual flame of inspiration.
  • No existence could be more tragic... and the only one at all comparable... is.. that of... Niels Henrik Abel, who died of consumption at twenty-six in 1829... just when Galois was ready to take the torch from his hand and to run... further. Abel had the inestimable privilege of living six years longer... full years at the time that genius was ripe... of divine inspiration. What would not Galois have given us, if he had been granted six more such years at the climax of his life.
  • [D]iscoverers of fundamental principles are not generally awarded much recompense. They often die misunderstood and unrewarded. But... the fame... of Galois ...is based upon the unlimited future. He well knew the pregnancy of his thoughts, yet they were even more far-reaching than he could possibly dream of.
  • His complete works fill only sixty-one small pages; but a French geometer, publishing a large volume some forty years after Galois' death, declared... it... simply a commentary on... [Galois'] discoveries. Since then, many more consequences have been deduced... and Galois'... ideas have influenced the whole of mathematical philosophy.
  • He had read... books of geometry as easily as a novel... No sooner had he begun to study algebra than he read Lagrange's original memoirs. This extraordinary facility had been at first a revelation... but... it became more difficult for him to curb his own domineering thought and to sacrifice it to the routine of class work. ...By 1827 it had reached a critical point. This might be called the second crisis of his childhood: his scientific initiation. His change of mood was observed by the family. Juvenile gaiety was suddenly replaced by concentration; his manners became stranger every day. A mad desire to march forward along the solitary path... possessed him.
  • In his last year at the college, 1828-1829... [his] teacher of mathematics... wrote of him: "This student has a marked superiority over all his school-mates. ...He works only at the highest parts of mathematics." ...[O]ther teachers were less indulgent. For physics and chemistry, the note often repeated was: "Very absent-minded, no work whatever."
  • [A]t the age of sixteen he believed that he had found a method of solving general equations of the fifth degree. ...[B]efore succeeding in proving the impossibility of such resolution, Abel had made the same mistake.
  • Galois was already trying to... enter the École polytechnique... as early as 1828 — but failed. This failure was very bitter to him... he considered it as unfair... [but] his extra knowledge could not compensate for his deficiencies... The next year he published his first paper, and sent his first communication to the Academie des Sciences... lost through Cauchy's negligence. This embittered Galois even more. A second failure to enter Polytechnique seemed to... climax... his misfortune...
  • On July 2... 1829, his father had been driven to commit suicide... This terrible blow, following many smaller miseries, left a very deep mark... His hatred of injustice became the more violent... his father's death incensed him, and developed his tendency to see injustice and baseness everywhere.
  • École polytechnique was... the highest mathematical school in France and... also a daughter of the Revolution who had remained faithful... The young Polytechnicians were the natural leaders of every political rebellion; liberalism was for them a matter of traditional duty. ...[T]hus twice sacred to Galois, and his failure to be accepted was a double misfortune.
  • In 1829 he entered the Ecole Normale... then passing through the most languid period of its existence. ...[T]here too, the main student body inclined toward liberalism, though their convictions were very weak and passive as compared with the... Polytechnique... Evariste suffered doubly, for his political desires were checked and his mathematical ability remained unrecognized.
  • His French biographer very clearly explains his attitude:

    There was in him a hardly disguised contempt for whosoever did not bow spontaneously and immediately before his superiority, a rebellion against a judgment which his conscience challenged beforehand and a sort of unhealthy pleasure in leading it further astray and in turning it entirely against himself. Indeed, it is frequently observed that those people who believe that they have most to complain of persecution could hardly do without it and, if need be, will provoke it. To pass oneself off for a fool is another way and not the least savory, of making fools of others.

    • Ref: René Taton, "L'œuvre mathématique de Galois" (1947) Revue d'histoire des sciences et de leurs applications
  • In the... ensuing year, he sent three more papers to mathematical journals and a new memoir to the Academie. The permanent secretary, Fourier, took it home with him, but died before having examined it, and... [it] was not retrieved... Thus his second memoir was lost like the former.
  • He considered himself a victim of a... social organization which... sacrifices genius to mediocrity, and... he cursed the hated regime of oppression which... precipitated his father's death and against which the storm was gathering.
  • Evariste was in the possession of his general principles by the beginning of 1830... at the age of eighteen, and that he... knew their importance. ...[H]e did not trouble himself to write his memoirs with sufficient clearness and to give the explanations... necessary because his thoughts were... novel. ...Instead ...Galois enveloped his thought in ...secrecy by his efforts to attain ...conciseness, that coquetry of mathematicians.
  • It was... another man who reentered the Ecole Normale in the autumn of 1830. ...The revolution had opened to him a fresh source of disillusion...
  • [U]nder a more liberal guise, the same oppression, the same favoritism, the same corruption soon took place under Louis-Philippe as under Charles X.
  • The publication in the "Gazette des Ecoles" of a letter of Galois... which... scornfully criticised the director's tergiversations was... the last of many offenses. On Dec. 9, he was invited to leave the school, and his expulsion was ratified by the Royal Council on Jan. 3, 1831.
  • To support himself Galois announced that he would give a private course of higher algebra... [A] new copy of his second lost memoir... communicated... to the Academie... was returned to him by Poisson, four months later, as being incomprehensible. Galois was partly responsible... for he had taken no pains to explain himself clearly. This was the last straw. ...[H]e plunged himself entirely into the political turmoil. ...He is said to have exclaimed: "If a corpse were needed to stir the people up, I would give mine."
  • May 9, 1831, at the end of a political banquet, being intoxicated—not with wine but with the ardent conversation of an evening—he proposed a sarcastic toast to the King. He held his glass and an open knife in one hand and said simply: "To Louis Philippe!" Of course he was soon arrested and sent to Ste. Pélagie. The lawyer persuaded him to maintain that he had actually said: "To Louis-Philippe, if he betray," and many witnesses affirmed... His attitude before the tribunal was ironical and provoking, yet the jury rendered a verdict of not proven and he was acquitted. On the following Fourteenth of July, the government... [had] him arrested as a preventive measure. He was given six months' imprisonment on the technical charge of carrying arms and wearing a military uniform, but he remained in Ste. Pélagie only until... sent to a convalescent home... A dreadful epidemic of cholera was then raging in Paris, and Galois' transfer had been determined by the poor state of his health.
  • He was now a prisoner on parole and took advantage... to carry on an intrigue with a woman... probably not... reputable ("une coquette de bas etage," says Raspail) .
  • [G]enius possessing... a mere boy, a fragile little body divided within itself by disproportionate forces, an undeveloped mind crushed mercilessly between the exaltation of scientific discovery and the exaltation of sentiment.
  • Four days later two men challenged him to a duel. ...According to Evariste's younger brother the duel was not fair. Evariste, weak as he was, had to deal with two ruffians hired to murder him.
  • The last letter addressed to... Auguste Chevalier, was a... scientific testament. Its seven pages, hastily written... contain a summary of the discoveries which he had been unable to develop. This statement is so concise and... full that its significance could be understood only gradually... It proves the depth of his insight, for it anticipates discoveries of a much later date. At the end of the letter, after requesting his friend to publish it and to ask Jacobi or Gauss to pronounce upon it, he added: "After that, I hope some people will find it profitable to unravel this mess. Je t'embrasse avec effusion." ...[T]he greatest mathematicians of the century have found it very profitable ...to clear up Galois' ideas.
    • Tr: I send you a warm kiss.
  • The duel took place on the 30th in the early morning, and he was grievously wounded by a shot in the abdomen. He was found by a peasant who transported him at 9:30 to the Hôpital Cochin. His younger brother... came and stayed with him, and as he was crying, Evariste tried to console him, saying: "Do not cry. I need all my courage to die at twenty." ...[H]e refused the assistance of a priest. ...[H]e breathed his last ...the following morning.
  • When genius... explodes suddenly, at the beginning and not at the end of life, or when we are at a loss to explain its... genesis, we can but feel that we are in the sacred presence of something vastly superior to talent. ...[I]t is not necessary to introduce any mystical idea, but it is one's duty to acknowledge the mystery. ...Galois' fateful existence helps one to understand Lowell's saying: "Talent is that which is in a man's power, genius is that in whose power man is." If Galois had been simply a mathematician of considerable ability, his life would have been far less tragic, for he could have used his mathematical talent for his own advancement and happiness; instead... the furor of mathematics—as one of his teachers said—possessed him and he had no alternative but absolute surrender to his destiny.
  • It is painful to think that a few rays of generosity from the heart of his elders might have saved this boy or... sweetened his life.
  • Galois... accomplished his task and... few men will... accomplish more. He has conquered the purest kind of immortality. As he wrote to his friends: "I take with me to the grave a conscience free from lie, free from patriot's blood". How many of the conventional heroes of history, how many of the kings, captains and statesmen could say the same?

History of Mathematics (1925)

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Vol. 2, by David Eugene Smith
  • Although the Greek use of continued fractions in the case of greatest common measure was well known in the Middle Ages, the modern theory of the subject may be said to have begun with Bombelli (1572). ...The first great memoir on the subject was Euler's De fractionibus continuis (1737), and in this work the foundation for the modern theory was laid. Among other interesting cases Euler developed e as a continued fraction, thus:
    Of the later contributors to the theory, special mention should be made of Lagrange (1767) and Galois.
    • Ref: Œuvres mathtmatiques d'Evariste Galois (1897) pp. 1-8.
  • The modern theory of equations in general is commonly said to date from Abel and Galois. The latter's posthumous (1846) memoir... established the theory... To him is due the discovery that to each equation there corresponds a group of substitutions (the "group of the equation") in which are reflected its essential characteristics. Galois' early death left without sufficient demonstration several important propositions, a gap which has since been filled.
  • The first formulas for the computation of the symmetric functions of the roots of an equation seem to have been worked out by Newton, although Girard (1629) had given, without proof, a formula for a power of the sum, and Cardan (1545) had made a slight beginning in the theory. In the 18th century Lagrange (1768) and Waring (1770, 1782) made several valuable contributions to the subject, but the first tables, reaching to the tenth degree, appeared in 1809 in the Meyer Hirsch Aufgabensammlung. In Cauchy's celebrated memoir on determinants (1812) the subject began to assume new prominence, and both he and Gauss (1816) made numerous and important additions to the theory. It is, however, since the discoveries by Galois that the subject has become one of great significance.

The Development Of Mathematics (1940)

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by E. T. Bell. See also 1945 edition.
  • Gauss took as the subject for his doctor’s dissertation (1799) a proof of the fundamental theorem of algebra: an algebraic equation has a root of the form a + bi, a , b real. ...After Girard’s conjecture, there had been attempts at proof, including essays by D’Alembert (1746) and Euler (1749). All were faulty, as were the first and fourth attempts (1799) by Gauss. ...[T]he fundamental theorem in its classic form, as proved in the theory of functions of a complex variable, is no longer regarded as belonging to algebra. It is supplanted in modern algebra by a statement which is almost a triviality. The basic ideas of the modern treatment go back to Galois.., Dedekind.., and Kronecker.., not to Gauss.
  • Three new approaches to number, in 1801 and in the 1830's, were to hint at the general concept of mathematical structure and reveal unsuspected horizons... That of 1801 was the concept of congruence, introduced by Gauss [age 24] in... the Disquisitiones arithmeticae... To this and the revolutionary work (1830-2) of E. Galois... in the theory of algebraic equations can be traced the partial execution of L. Kronecker’s... revolutionary program in the 1880's for basing all mathematics on the natural numbers.
  • We must... indicate the involuntary participation of E. Galois... and N. H. Abel... in the development of Kronecker’s Pythagoreanism. Galois... adhered to no such creed. Nor did Abel. But it was in the attempt to understand and elucidate the Galois theory of equations, left (1832)... in a... fragmentary and unapproachable condition, that Kronecker acquired some of his skill. Both Kronecker and Dedekind, two of the founders (E. E. Kummer... being a third) of the theory of algebraic numbers, were inspired partly by their scrutiny of the Galois theory to begin their... revolutionary work in algebra and arithmetic. ...Galois and Abel mark the beginning of one modern approach to algebra. The transition from highly finished individual theorems to abstract and widely inclusive theories...
  • Younger than Gauss by thirty-four years, and dying twenty-three years before him, Galois... seems more modern... Gauss terminated his investigations on the nature of the solutions of algebraic equations with the binomial equation
    Galois grasped and solved (1830) the general problem, proving, among other things, necessary and sufficient conditions for the solution by radicals of any algebraic equation. Mathematics after Gauss, and partly during his own lifetime, became more general and more abstract... Interest in special problems sharply declined if there was a general problem including the special instances to be attacked... [i.e.,] mathematics after Gauss turned to the construction of inclusive theories and general methods which, theoretically... implied... detailed solutions of infinities of special problems. In this sense Galois was more modern...
  • Applications, or developments, of... extensions of number followed two main directions. ...The second, in the arithmetical spirit of Gauss, guided in part by the abstract algebraic outlook of Galois, led to a partial but extensive arithmetization of algebra.
  • Of all these influences, two in particular are germane... the development of linear algebra; and the infiltration of the ideas of Abel and Galois into algebra as a whole. The Galois theory of equations was acknowledged by both Dedekind and Kronecker to be the inspiration for their own general and semi-arithmetical approach to algebra. Two of the basic concepts of the Galois theory, domains of rationality, or fields, and groups, were the point of departure. ...Common algebra is the most familiar example of a field.
  • The earliest recognitions of fields, but without explicit definition, appear to be in the researches of Abel (1828) and Galois (1830-1) on the solution of equations by radicals. The first formal lectures on the Galois theory were those of Dedekind to two students in the early 1850's. Kronecker also at that time began his studies on abelian equations. It appears that the concept of a field passed into mathematics through the arithmetical works of Dedekind and Kronecker.
  • Galois... made the terminal contribution so far as algebraic equations are concerned, and subsequent reworkings of his initial theory have added nothing basically new to his criteria for solvability by radicals. Even the modernized presentation of the Galois theory, as in the streamlined model of E. Artin... is a tribute to the mathematical creed of Galois, in its elimination of all superfluous machinery. For this modern release from algebraic calculation, the direct approach of A. E. ‘Emmy’ Noether... in the 1920’s was primarily responsible. Much of her mathematics was in the spirit of Galois. But his methods, sharpened and generalized by his successors, have transcended the problem for which they were invented, and have rejuvenated much of living pure mathematics.
  • [C]oncerning the vital residue of the theory of algebraic numbers... it, like the Galois theory, can be traced to definite, highly special problems. Neither Galois nor the creators of the theory of algebraic number fields set out deliberately to revolutionize a mathematical technique; their comprehensive methods were invented to solve specific problems.
    Such appears to have been the usual path to abstractness, generality, and increased power.     Some difficult problem... is taken as the point of departure without any conscious effort to create a comprehensive theory; repeated failures to achieve a solution by known procedures force the invention of new methods; and finally, the new methods, having been necessitated by a problem which appeared in the historical development, themselves pass into the main stream.
  • The Galois theory of equations itself was the concluding episode in about three centuries of effort to penetrate the arithmetical nature of the roots of algebraic equations.
  • The work of Galois and his successors showed that the nature, or explicit definition, of the roots of an algebraic equation is reflected in the structure of the group of the equation for the field of its coefficients. This group can be determined nontentatively in a finite number of steps, although, as Galois himself emphasized, his theory is not intended to be a practical method for solving equations. But, as stated by Hilbert, the Galois theory and the theory of algebraic numbers have their common root in that of algebraic fields. The last was initiated by Galois, developed by Dedekind and Kronecker in the mid-nineteenth century, refined and extended in the late nineteenth century by Hilbert and others, and finally, in the twentieth century, given a new direction by the work of Steinitz in 1910, and in that of E. Noether and her school since 1920.
  • Even before Galois coined the term 'group,' A. L. Cauchy... made (1815) extensive investigations in what are now called permutation groups, and discovered some of the simpler basic theorems.
  • [A]ny mathematician today must be impressed by the apparent permanence of the ideas introduced by Abel... and Galois.., and the profound difference between their approach to mathematics and that of their predecessors including, in some respects, Gauss... To these young men, perhaps more than to any other two mathematicians, can be traced the pursuit of generality which distinguishes the mathematics of the recent period, beginning with Gauss in 1801, from that of the middle period. They initiated... the deliberate search for inclusive methods and comprehensive theories. Their forerunners in the middle period were Descartes with his general method in geometry; Newton and Leibniz with the differential and integral calculus created to attack the mathematics of continuity by a uniform procedure; and Lagrange, with his universal method in mechanics. Their contemporary in recent mathematics was Gauss, who in his arithmetic sought to unify much of the uncorrelated work of the leading arithmeticians from Fermat to Euler, Lagrange, and Legendre. Both Abel and Galois acknowledged their indebtedness to the theory of cyclotomy created by Gauss; and although they went far beyond him in their own algebra (Abel in analysis also), it is at least conceivable that neither Abel nor Galois would have chosen the road he followed had it not been for the hints in the Gaussian theory of binomial equations.
  • Both Abel and Galois died long before their time, Abel at the age of twenty-seven from tuberculosis induced by poverty, Galois at twenty-one of a pistol shot received in a meaningless duel. When Abel’s genius was recognized, he was subsidized by friends and the Norwegian government. By nature he was genial and optimistic. Galois spent a considerable part of his five or six productive years in a hopeless fight against the stupidities and malicious jealousy of teachers and the smug indifference of academicians. Not at first quarrelsome or perverse, he became both.
  • Whoever, if anybody, was responsible for the colossal waste represented by these two premature deaths, it seems probable that mathematics was needlessly deprived of the natural successsors of Gauss. What Abel and Galois might have accomplished in a normal lifetime cannot be even conjectured. ...Early maturity and sustained productivity are the rule, not the exception, for the greatest mathematicians. It may be true that the most original ideas come early; but it takes time to work them out.
  • Gauss, the penniless son of a day laborer, was educated by society as represented by the Duke of Brunswick. Today he would be educated at public expense...
    An Abel, no doubt, would be sent by the municipal health authorities to a sanitarium, where he might recover.
    A Galois... would find himself at outs with respectability, or in the protective custody of the police on some trumped-up charge... or in a concentration camp. For there is but little evidence that teachers are less helpless in the disturbing presence of a mind of the very highest intelligence than they were... or that the guardians of law and order are less nervous than they were when they sentenced Galois to... jail on a legal technicality. Aesop’s fable of the peacock and the crows has an element of permanence... you are different from us; get out or be plucked.
  • Congruences were responsible for one theory of far more than merely arithmetical interest. The notation for a congruence suggests the introduction of appropriate 'imaginaries' to supply the congruence with roots equal in number to the degree of the congruence when there is a deficiency of real roots. As in the corresponding algebraic problem, it is not obvious that imaginaries can be introduced consistently. That they can, was first proved in 1830 by Galois, who invented the required 'numbers,' since called Galois imaginaries, for the solution of any irreducible congruence F(x) = 0 mod p , where p is prime. He thus obtained a generalization of Fermat’s theorem, and laid the foundation of the theory of finite fields. ...Galois was eighteen when he invented his imaginaries.
  • Probably almost anyone who has ever seriously attempted to solve differential equations by the Lie theory will appreciate the labor inherent in any such heroic project as Wilczynski’s and agree with Galois that, whatever the nature of its unchallenged merits, the theory of groups does not afford a practicable method for solving equations. Galois of course was speaking of algebraic equations, but his opinion, in the judgment of experts in the Lie theory, carries over to differential equations. Beyond a not very advanced stage of complexity, the calculations become prohibitive to even the most persevering obstinacy.
  • [A]s noted by Lie in the grand summary (1893) of his lifework, the nineteenth century’s greatest effort in formal algebra—as distinguished from the more abstract, structural algebra originating with Galois—was absorbed in analysis.
  • Galois, who was Lie's idol, indirectly inspired the application of continuous groups to differential equations. In a letter of 1874 to A. Mayer, Lie observes that "In the theory of algebraic equations before Galois only these questions were proposed: Is an equation solvable by radicals, and how is it to be solved? Since Galois, among other questions proposed is this: How is an equation to be solved by radicals in the simplest way possible? I believe the time is come to make a similar progress in differential equations."
  • Just as the algebraic theory characterizes the nature of the irrationalities required for the solution of a given algebraic equation, so does the structural theory of differential equations characterize and classify the functions defined by a system of differential equations. ...[t]he initial impulse for a structural theory came from Lie's transformation groups. In spite of Picard's deprecatory estimate of his own contribution, which, historically, inaugurated the project, as "only a very natural extension to an analytic problem of the extremely fruitful ideas introduced into algebra by Galois," the problem of devising a structural theory for differential equations was no facile exercise in principles already classic.
  • Although... structural theories of a major division of analysis originated in the late nineteenth century, they are more in the spirit of the general analysis of the twentieth. Their primary objectives are to discover what can be done rather than to do it, and to give criteria for what cannot be done. ...[A]s in Abel’s proof that the general quintic is not solvable by radicals, a demonstration of impossibility definitely disposed of what might seem a reasonable problem. Once more the methodology of Abel and Galois made an outstanding contribution to the development of mathematics. In this connection it is interesting to recall Lie’s opinion that the pattern of nineteenth century mathematics was laid out by four men, Gauss, Cauchy, Abel, and Galois.
  • With this highly abstract definition of a space in mind, we return once more to Klein’s program and its successors. It is interesting to observe the abstract identity between the following description of spacial structure and structure as described in connection with modern algebra, and further to note once more that the basic concepts originated with Galois. Two spaces are called equivalent or (simply) isomorphic if there is a one-one correspondence between the objects in the spaces which establishes a one-one correspondence between all the properties constituting the structures of the respective spaces. When this is applied to two spaces which are the same, there is thus defined what is called an automorphism of the space. It follows... from these definitions that all the automorphisms of a given space form a group.
  • Galois... had nothing of the topologist about him...

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