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Évariste Galois – Wikipedia

Évariste Galois – Wikipedia

2023-11-27 14:19:21

French mathematician (1811–1832)

Évariste Galois (;[1] French: [evaʁist ɡalwa]; 25 October 1811 – 31 Might 1832) was a French mathematician and political activist. Whereas nonetheless in his teenagers, he was capable of decide a necessary and sufficient condition for a polynomial to be solvable by radicals, thereby fixing an issue that had been open for 350 years. His work laid the foundations for Galois theory and group theory,[2] two main branches of abstract algebra.

Galois was a staunch republican and was closely concerned within the political turmoil that surrounded the French Revolution of 1830. On account of his political activism, he was arrested repeatedly, serving one jail sentence of a number of months. For causes that stay obscure, shortly after his launch from jail, Galois fought in a duel and died of the injuries he suffered.[3]

Formative years[edit]

Galois was born on 25 October 1811 to Nicolas-Gabriel Galois and Adélaïde-Marie (née Demante).[2][4] His father was a Republican and was head of Bourg-la-Reine’s liberal party. His father grew to become mayor of the village[2] after Louis XVIII returned to the throne in 1814. His mom, the daughter of a jurist, was a fluent reader of Latin and classical literature and was answerable for her son’s training for his first twelve years.

The Cour d’honneur of the Lycée Louis-le-Grand, which Galois attended as a boy.

In October 1823, he entered the Lycée Louis-le-Grand the place his trainer Louis Paul Émile Richard acknowledged his brilliance.[5] On the age of 14, he started to take a severe curiosity in mathematics.[5]

Galois discovered a replica of Adrien-Marie Legendre‘s Éléments de Géométrie, which, it’s mentioned, he learn “like a novel” and mastered on the first studying. At 15, he was studying the unique papers of Joseph-Louis Lagrange, such because the Réflexions sur la résolution algébrique des équations which doubtless motivated his later work on equation concept,[6] and Leçons sur le calcul des fonctions, work supposed for skilled mathematicians, but his classwork remained uninspired and his academics accused him of placing on the airs of a genius.[4]

Budding mathematician[edit]

In 1828, Galois tried the doorway examination for the École Polytechnique, essentially the most prestigious establishment for arithmetic in France on the time, with out the same old preparation in arithmetic, and failed for lack of explanations on the oral examination. In that very same yr, he entered the École Normale (then often called l’École préparatoire), a far inferior establishment for mathematical research at the moment, the place he discovered some professors sympathetic to him.[citation needed]

Augustin-Louis Cauchy reviewed Galois’s early mathematical papers.

Within the following yr Galois’s first paper, on continued fractions,[7]
was revealed. It was at across the identical time that he started making elementary discoveries within the concept of polynomial equations. He submitted two papers on this matter to the Academy of Sciences. Augustin-Louis Cauchy refereed these papers, however refused to simply accept them for publication for causes that also stay unclear. Nonetheless, despite many claims on the contrary, it’s extensively held that Cauchy acknowledged the significance of Galois’s work, and that he merely instructed combining the 2 papers into one with the intention to enter it within the competitors for the Academy’s Grand Prize in Arithmetic. Cauchy, an eminent mathematician of the time although with political beliefs that had been diametrically against these of Galois, thought-about Galois’s work to be a possible winner.[8]

On 28 July 1829, Galois’s father died by suicide after a bitter political dispute with the village priest.[9] A few days later, Galois made his second and final try and enter the Polytechnique and failed but once more.[9] It’s undisputed that Galois was greater than certified; accounts differ on why he failed. Extra believable accounts state that Galois made too many logical leaps and baffled the incompetent examiner, which enraged Galois. The latest loss of life of his father might have additionally influenced his habits.[4]

Having been denied admission to the École polytechnique, Galois took the Baccalaureate examinations with the intention to enter the École normale.[9] He handed, receiving his diploma on 29 December 1829.[9] His examiner in arithmetic reported, “This pupil is usually obscure in expressing his concepts, however he’s clever and exhibits a outstanding spirit of analysis.”

Galois submitted his memoir on equation concept a number of instances, nevertheless it was by no means revealed in his lifetime. Although his first try was refused by Cauchy, in February 1830 following Cauchy’s suggestion he submitted it to the Academy’s secretary Joseph Fourier,[9] to be thought-about for the Grand Prix of the Academy. Sadly, Fourier died quickly after,[9] and the memoir was misplaced.[9] The prize can be awarded that yr to Niels Henrik Abel posthumously and in addition to Carl Gustav Jacob Jacobi. Regardless of the misplaced memoir, Galois revealed three papers that yr. One laid the foundations for Galois theory.[10] The second was in regards to the numerical decision of equations (root finding in trendy terminology).[11] The third was an necessary one in number theory, wherein the idea of a finite field was first articulated.[12]

Political firebrand[edit]

Battle for the City Corridor by Jean-Victor Schnetz. Galois, as a staunch republican, would have needed to take part within the July Revolution of 1830 however was prevented by the director of the École Normale.

Galois lived throughout a time of political turmoil in France. Charles X had succeeded Louis XVIII in 1824, however in 1827 his party suffered a major electoral setback and by 1830 the opposition liberal party became the majority. Charles, confronted with political opposition from the chambers, staged a coup d’état, and issued his infamous July Ordinances, touching off the July Revolution[9] which ended with Louis Philippe changing into king. Whereas their counterparts on the Polytechnique had been making historical past within the streets, Galois, on the École Normale, was locked in by the varsity’s director. Galois was incensed and wrote a blistering letter criticizing the director, which he submitted to the Gazette des Écoles, signing the letter along with his full identify. Though the Gazettes editor omitted the signature for publication, Galois was expelled.[13]

Though his expulsion would have formally taken impact on 4 January 1831, Galois give up faculty instantly and joined the staunchly Republican artillery unit of the National Guard. He divided his time between his mathematical work and his political affiliations. Attributable to controversy surrounding the unit, quickly after Galois grew to become a member, on 31 December 1830, the artillery of the Nationwide Guard was disbanded out of worry that they may destabilize the federal government. At across the identical time, nineteen officers of Galois’s former unit had been arrested and charged with conspiracy to overthrow the federal government.

In April 1831, the officers had been acquitted of all fees, and on 9 Might 1831, a banquet was held of their honor, with many illustrious individuals current, akin to Alexandre Dumas. The proceedings grew riotous. In some unspecified time in the future, Galois stood and proposed a toast wherein he mentioned, “To Louis Philippe,” with a dagger above his cup. The republicans on the banquet interpreted Galois’s toast as a risk towards the king’s life and cheered. He was arrested the next day at his mom’s home and held in detention at Sainte-Pélagie prison till 15 June 1831, when he had his trial.[8] Galois’s protection lawyer cleverly claimed that Galois really mentioned, “To Louis-Philippe, if he betrays,” however that the qualifier was drowned out within the cheers. The prosecutor requested a couple of extra questions, and maybe influenced by Galois’s youth, the jury acquitted him that very same day.[8][9][13][14]

On the next Bastille Day (14 July 1831), Galois was on the head of a protest, sporting the uniform of the disbanded artillery, and got here closely armed with a number of pistols, a loaded rifle, and a dagger. He was once more arrested.[9] Throughout his keep in jail, Galois at one level drank alcohol for the primary time on the goading of his fellow inmates. One in every of these inmates, François-Vincent Raspail, recorded what Galois mentioned whereas drunk in a letter from 25 July. Excerpted from the letter:[8]

And I let you know, I’ll die in a duel on the event of some coquette de bas étage. Why? As a result of she’s going to invite me to avenge her honor which one other has compromised.
Have you learnt what I lack, my pal? I can confide it solely to you: it’s somebody whom I can love and love solely in spirit. I’ve misplaced my father and nobody has ever changed him, do you hear me…?

Raspail continues that Galois, nonetheless in a delirium, tried suicide, and that he would have succeeded if his fellow inmates hadn’t forcibly stopped him.[8] Months later, when Galois’s trial occurred on 23 October, he was sentenced to 6 months in jail for illegally sporting a uniform.[9][15][16] Whereas in jail, he continued to develop his mathematical concepts. He was launched on 29 April 1832.

Ultimate days[edit]

Siméon Denis Poisson reviewed Galois’s paper on equation concept and declared it “incomprehensible”.

Galois returned to arithmetic after his expulsion from the École Normale, though he continued to spend time in political actions. After his expulsion grew to become official in January 1831, he tried to begin a non-public class in superior algebra which attracted some curiosity, however this waned, because it appeared that his political activism had precedence.[4][8] Siméon Denis Poisson requested him to submit his work on the theory of equations, which he did on 17 January 1831. Round 4 July 1831, Poisson declared Galois’s work “incomprehensible”, declaring that “[Galois’s] argument is neither sufficiently clear nor sufficiently developed to permit us to guage its rigor”; nevertheless, the rejection report ends on an encouraging notice: “We might then recommend that the writer ought to publish the entire of his work with the intention to type a definitive opinion.”[17]
Whereas Poisson’s report was made earlier than Galois’s 14 July arrest, it took till October to succeed in Galois in jail. It’s unsurprising, within the gentle of his character and state of affairs on the time, that Galois reacted violently to the rejection letter, and determined to desert publishing his papers by way of the Academy and as a substitute publish them privately by way of his pal Auguste Chevalier. Apparently, nevertheless, Galois didn’t ignore Poisson’s recommendation, as he started amassing all his mathematical manuscripts whereas nonetheless in jail, and continued sharpening his concepts till his launch on 29 April 1832,[13] after which he was someway talked right into a duel.[9]

Galois’s deadly duel occurred on 30 Might.[18] The true motives behind the duel are obscure. There was a lot hypothesis about them. What is thought is that, 5 days earlier than his loss of life, he wrote a letter to Chevalier which clearly alludes to a damaged love affair.[8]

Some archival investigation on the unique letters means that the girl of romantic curiosity was Stéphanie-Félicie Poterin du Motel,[19] the daughter of the doctor on the hostel the place Galois stayed over the past months of his life. Fragments of letters from her, copied by Galois himself (with many parts, akin to her identify, both obliterated or intentionally omitted), can be found.[20] The letters trace that du Motel had confided a few of her troubles to Galois, and this may need prompted him to impress the duel himself on her behalf. This conjecture can also be supported by different letters Galois later wrote to his associates the night time earlier than he died. Galois’s cousin, Gabriel Demante, when requested if he knew the reason for the duel, talked about that Galois “discovered himself within the presence of a supposed uncle and a supposed fiancé, every of whom provoked the duel.” Galois himself exclaimed: “I’m the sufferer of an notorious coquette and her two dupes.”[13]

Far more detailed hypothesis primarily based on these scant historic particulars has been interpolated by lots of Galois’s biographers, such because the steadily repeated hypothesis that your entire incident was stage-managed by the police and royalist factions to get rid of a political enemy.[citation needed]

As to his opponent within the duel, Alexandre Dumas names Pescheux d’Herbinville,[14] who was really one of many nineteen artillery officers whose acquittal was celebrated on the banquet that occasioned Galois’s first arrest.[21] Nonetheless, Dumas is alone on this assertion, and if he had been appropriate it’s unclear why d’Herbinville would have been concerned. It has been speculated that he was du Motel’s “supposed fiancé” on the time (she finally married another person), however no clear proof has been discovered supporting this conjecture. Then again, extant newspaper clippings from only some days after the duel give an outline of his opponent (recognized by the initials “L.D.”) that seem to extra precisely apply to one in all Galois’s Republican associates, likely Ernest Duchatelet, who was imprisoned with Galois on the identical fees.[22] Given the conflicting info obtainable, the true id of his killer could be misplaced to historical past.

Regardless of the causes behind the duel, Galois was so satisfied of his impending loss of life that he stayed up all night time writing letters to his Republican associates and composing what would develop into his mathematical testomony, the well-known letter to Auguste Chevalier outlining his concepts, and three hooked up manuscripts.[23] Mathematician Hermann Weyl mentioned of this testomony, “This letter, if judged by the novelty and profundity of concepts it accommodates, is maybe essentially the most substantial piece of writing in the entire literature of mankind.” Nonetheless, the legend of Galois pouring his mathematical ideas onto paper the night time earlier than he died appears to have been exaggerated.[8] In these last papers, he outlined the tough edges of some work he had been doing in evaluation and annotated a replica of the manuscript submitted to the Academy and different papers.

The Galois memorial within the cemetery of Bourg-la-Reine. Évariste Galois was buried in a standard grave and the precise location is unknown.

Early within the morning of 30 Might 1832, he was shot within the abdomen,[18] was deserted by his opponents and his personal seconds, and was discovered by a passing farmer. He died the next morning[18] at ten o’clock within the Hôpital Cochin (in all probability of peritonitis), after refusing the places of work of a priest. His funeral resulted in riots.[18] There have been plans to provoke an rebellion throughout his funeral, however throughout the identical time the leaders heard of Basic Jean Maximilien Lamarque‘s loss of life and the rising was postponed with none rebellion occurring till 5 June. Solely Galois’s youthful brother was notified of the occasions previous to Galois’s loss of life.[24] Galois was 20 years previous. His last words to his youthful brother Alfred had been:

“Ne pleure pas, Alfred ! J’ai besoin de tout mon braveness pour mourir à vingt ans !”
(Do not weep, Alfred! I would like all my braveness to die at twenty!)

On 2 June, Évariste Galois was buried in a standard grave of the Montparnasse Cemetery whose actual location is unknown.[18][16] Within the cemetery of his native city – Bourg-la-Reine – a cenotaph in his honour was erected beside the graves of his family.[25]

Évariste Galois died in 1832. Joseph Liouville started learning Galois’s unpublished papers in 1842 and acknowledged their worth in 1843. It isn’t clear what occurred within the 10 years between 1832 and 1842 nor what finally impressed Joseph Liouville to start studying Galois’s papers. Jesper Lützen explores this topic at some size in Chapter XIV Galois Idea of his e book about Joseph Liouville with out reaching any definitive conclusions.[26]

It’s actually attainable that mathematicians (together with Liouville) didn’t need to publicize Galois’s papers as a result of Galois was a republican political activist who died 5 days earlier than the June Rebellion, an unsuccessful anti-monarchist revolt of Parisian republicans. In Galois’s obituary, his pal Auguste Chevalier virtually accused academicians on the École Polytechnique of getting killed Galois since, if that they had not rejected his work, he would have develop into a mathematician and wouldn’t have devoted himself to the republican political activism for which some believed he was killed.[26]

Provided that France was nonetheless residing within the shadow of the Reign of Terror and the Napoleonic era, Liouville may need waited till the June Rebellion‘s political turmoil subsided earlier than turning his consideration to Galois’s papers.[26]

Liouville lastly revealed Galois’s manuscripts within the October–November 1846 problem of the Journal de Mathématiques Pures et Appliquées.[27][28] Galois’s most well-known contribution was a novel proof that there isn’t any quintic formula – that’s, that fifth and better diploma equations will not be usually solvable by radicals. Though Niels Henrik Abel had already proved the impossibility of a “quintic formula” by radicals in 1824 and Paolo Ruffini had revealed an answer in 1799 that turned out to be flawed, Galois’s strategies led to deeper analysis into what’s now referred to as Galois Theory, which can be utilized to find out, for any polynomial equation, whether or not it has an answer by radicals.

Contributions to arithmetic[edit]

The ultimate web page of Galois’s mathematical testomony, in his personal hand. The phrase “to decipher all this mess” (“déchiffrer tout ce gâchis”) is on the second to the final line.

From the closing traces of a letter from Galois to his pal Auguste Chevalier, dated 29 Might 1832, two days earlier than Galois’s loss of life:[23]

See Also

Tu prieras publiquement Jacobi ou Gauss de donner leur avis, non sur la vérité, mais sur l’significance des théorèmes.

Après cela, il y aura, j’espère, des gens qui trouveront leur revenue à déchiffrer tout ce gâchis.

(Ask Jacobi or Gauss publicly to provide their opinion, not as to the reality, however as to the significance of those theorems. Later there might be, I hope, some individuals who will discover it to their benefit to decipher all this mess.)

Throughout the 60 or so pages of Galois’s collected works are many necessary concepts which have had far-reaching penalties for almost all branches of arithmetic.[29][30]
His work has been in comparison with that of Niels Henrik Abel (1802–1829), a recent mathematician who additionally died at a really younger age, and far of their work had important overlap.


Whereas many mathematicians earlier than Galois gave consideration to what are actually often called groups, it was Galois who was the primary to make use of the phrase group (in French groupe) in a way near the technical sense that’s understood at the moment, making him among the many founders of the department of algebra often called group theory. He referred to as the decomposition of a gaggle into its left and proper cosets a correct decomposition if the left and proper cosets coincide, which is what at the moment is named a standard subgroup.[23] He additionally launched the idea of a finite field (also referred to as a Galois field in his honor) in primarily the identical type as it’s understood at the moment.[12]

In his final letter to Chevalier[23] and hooked up manuscripts, the second of three, he made primary research of linear teams over finite fields:

Galois concept[edit]

Galois’s most important contribution to arithmetic is his improvement of Galois concept. He realized that the algebraic answer to a polynomial equation is expounded to the construction of a gaggle of permutations related to the roots of the polynomial, the Galois group of the polynomial. He discovered that an equation might be solved in radicals if one can discover a sequence of subgroups of its Galois group, each regular in its successor with abelian quotient, that’s, its Galois group is solvable. This proved to be a fertile strategy, which later mathematicians tailored to many different fields of arithmetic apart from the theory of equations to which Galois initially utilized it.[29]


Galois additionally made some contributions to the idea of Abelian integrals and continued fractions.

As written in his final letter,[23] Galois handed from the examine of elliptic capabilities to consideration of the integrals of essentially the most normal algebraic differentials, at the moment referred to as Abelian integrals. He labeled these integrals into three classes.

Continued fractions[edit]

In his first paper in 1828,[7] Galois proved that the common continued fraction which represents a quadratic surd ζ is solely periodic if and provided that ζ is a reduced surd, that’s, and its conjugate
satisfies .

In reality, Galois confirmed greater than this. He additionally proved that if ζ is a decreased quadratic surd and η is its conjugate, then the continued fractions for ζ and for (−1/η) are each purely periodic, and the repeating block in a type of continued fractions is the mirror picture of the repeating block within the different. In symbols now we have

the place ζ is any decreased quadratic surd, and η is its conjugate.

From these two theorems of Galois a end result already identified to Lagrange will be deduced. If r > 1 is a rational quantity that isn’t an ideal sq., then

Particularly, if n is any non-square optimistic integer, the common continued fraction enlargement of √n accommodates a repeating block of size m, wherein the primary m − 1 partial denominators type a palindromic string.

See additionally[edit]

  1. ^ “Galois theory”. Random House Webster’s Unabridged Dictionary.
  2. ^ a b c C., Bruno, Leonard (c. 2003) [1999]. Math and mathematicians : the history of math discoveries around the world. Baker, Lawrence W. Detroit, Mich.: U X L. p. 171. ISBN 978-0787638139. OCLC 41497065.{{cite book}}: CS1 maint: a number of names: authors checklist (link)
  3. ^ C., Bruno, Leonard (2003) [1999]. Math and mathematicians : the history of math discoveries around the world. Baker, Lawrence W. Detroit, Mich.: U X L. pp. 171, 174. ISBN 978-0787638139. OCLC 41497065.{{cite book}}: CS1 maint: a number of names: authors checklist (link)
  4. ^ a b c d Stewart, Ian (1973). Galois Theory. London: Chapman and Corridor. pp. xvii–xxii. ISBN 978-0-412-10800-6.
  5. ^ a b C., Bruno, Leonard (2003) [1999]. Math and mathematicians : the history of math discoveries around the world. Baker, Lawrence W. Detroit, Mich.: U X L. p. 172. ISBN 978-0787638139. OCLC 41497065.{{cite book}}: CS1 maint: a number of names: authors checklist (link)
  6. ^ “Réflexions sur la résolution algébrique des équations”. britannica encyclopedia.
  7. ^ a b Galois, Évariste (1828). “Démonstration d’un théorème sur les fractions continues périodiques”. Annales de Mathématiques. XIX: 294.
  8. ^ a b c d e f g h Rothman, Tony (1982). “Genius and Biographers: The Fictionalization of Evariste Galois”. The American Mathematical Month-to-month. 89 (2): 84–106. doi:10.2307/2320923. JSTOR 2320923. Retrieved 31 January 2015.
  9. ^ a b c d e f g h i j k l C., Bruno, Leonard (2003) [1999]. Math and mathematicians : the history of math discoveries around the world. Baker, Lawrence W. Detroit, Mich.: U X L. p. 173. ISBN 978-0787638139. OCLC 41497065.{{cite book}}: CS1 maint: a number of names: authors checklist (link)
  10. ^ Galois, Évariste (1830). “Analyse d’un Mémoire sur la résolution algébrique des équations”. Bulletin des Sciences Mathématiques. XIII: 271.
  11. ^ Galois, Évariste (1830). “Notice sur la résolution des équations numériques”. Bulletin des Sciences Mathématiques. XIII: 413.
  12. ^ a b Galois, Évariste (1830). “Sur la théorie des nombres”. Bulletin des Sciences Mathématiques. XIII: 428.
  13. ^ a b c d Dupuy, Paul (1896). “La vie d’Évariste Galois”. Annales Scientifiques de l’École Normale Supérieure. 13: 197–266. doi:10.24033/asens.427.
  14. ^ a b Dumas (père), Alexandre. “CCIV”. Mes Mémoires. ISBN 978-1-4371-5595-2. Retrieved 13 April 2010.
  15. ^ Bell, Eric Temple (1986). Males of Arithmetic. New York: Simon and Schuster. ISBN 978-0-671-62818-5.
  16. ^ a b Escofier, Jean-Pierre (2001). Galois Theory. Springer. pp. 222–224. ISBN 978-0-387-98765-1.
  17. ^ Taton, R. (1947). “Les relations d’Évariste Galois avec les mathématiciens de son temps”. Revue d’Histoire des Sciences et de Leurs Functions. 1 (2): 114–130. doi:10.3406/rhs.1947.2607.
  18. ^ a b c d e C., Bruno, Leonard (2003) [1999]. Math and mathematicians : the history of math discoveries around the world. Baker, Lawrence W. Detroit, Mich.: U X L. p. 174. ISBN 978-0787638139. OCLC 41497065.{{cite book}}: CS1 maint: a number of names: authors checklist (link)
  19. ^ Infantozzi, Carlos Alberti (1968). “Sur la mort d’Évariste Galois”. Revue d’Histoire des Sciences et de Leurs Functions. 21 (2): 157. doi:10.3406/rhs.1968.2554.
  20. ^ Bourgne, R.; J.-P. Azra (1962). Écrits et mémoires mathématiques d’Évariste Galois. Paris: Gauthier-Villars.
  21. ^ Blanc, Louis (1844). The History of Ten Years, 1830–1840, Volume 1. London: Chapman and Corridor. p. 431.
  22. ^ Dalmas, Andre (1956). Évariste Galois: Révolutionnaire et Géomètre. Paris: Fasquelle.
  23. ^ a b c d e Galois, Évariste (1846). “Lettre de Galois à M. Auguste Chevalier”. Journal de Mathématiques Pures et Appliquées. XI: 408–415. Retrieved 4 February 2009.
  24. ^ Coutinho, S.C. (1999). The Mathematics of Ciphers. Natick: A Ok Peters, Ltd. pp. 127–128. ISBN 978-1-56881-082-9.
  25. ^ Toti Rigatelli, Laura (1996). Evariste Galois, 1811–1832 (Vita mathematica, 11). Birkhäuser. p. 114. ISBN 978-3-7643-5410-7.
  26. ^ a b c Lützen, Jesper (1990). “Chapter XIV: Galois Idea”. Joseph Liouville 1809–1882: Grasp of Pure and Utilized Arithmetic. Research within the Historical past of Arithmetic and Bodily Sciences. Vol. 15. Springer-Verlag. pp. 559–580. ISBN 3-540-97180-7.
  27. ^ Galois, Évariste (1846). “OEuvres mathématiques d’Évariste Galois”. Journal de Mathématiques Pures et Appliquées. XI: 381–444. Retrieved 4 February 2009.
  28. ^ Pierpont, James (1899). “Review: Oeuvres mathématiques d’Evariste Galois; publiées sous les auspices de la Société Mathématique de France, avec une introduction par M. EMILE PICARD. Paris, Gauthier-Villars et Fils, 1897. 8vo, x + 63 pp” (PDF). Bull. Amer. Math. Soc. 5 (6): 296–300. doi:10.1090/S0002-9904-1899-00599-8. In 1897 the French Mathematical Society reprinted the 1846 publication.
  29. ^ a b Lie, Sophus (1895). “Affect de Galois sur le Développement des Mathématiques”. Le centenaire de l’École Normale 1795–1895. Hachette.
  30. ^ See additionally: Sophus Lie, “Influence de Galois sur le développement des mathématiques” in: Évariste Galois, Oeuvres Mathématiques publiées en 1846 dans le Journal de Liouville (Sceaux, France: Éditions Jacques Gabay, 1989), appendix pages 1–9.
  31. ^ Letter, p. 410
  32. ^ Letter, p. 411
  33. ^ Wilson, Robert A. (2009). “Chapter 1: Introduction”. The finite easy teams. Graduate Texts in Mathematics 251. Vol. 251. Berlin, New York: Springer-Verlag. doi:10.1007/978-1-84800-988-2. ISBN 978-1-84800-987-5. Zbl 1203.20012, 2007 preprint CS1 maint: postscript (link)
  34. ^ Letter, pp. 411–412
  35. ^ “Galois’s last letter, translated” (PDF).


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