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Évariste Galois

"Galois" redirects here. For other uses, see Gallois (disambiguation).

Évariste Galois (; French: ; 25 October 1811 – 31 May 1832) was a French mathematician and political activist. While still in his teens, he was able to determine a necessary and sufficient condition for a polynomial to be solvable by radicals, thereby solving a problem that had been open for 350 years. His work laid the foundations for Galois theory and group theory, two major branches of abstract algebra. He was a staunch republican and was heavily involved in the political turmoil that surrounded the French Revolution of 1830. As a result of his political activism, he was arrested repeatedly, serving one jail sentence of several months. For reasons that remain obscure, shortly after his release from prison, he fought in a duel, and died of the wounds he suffered.

Évariste Galois
A portrait of Évariste Galois aged about 15
Born(1811-10-25)25 October 1811
Died31 May 1832(1832-05-31) (aged 20)
Alma materÉcole préparatoire
Known forWork on theory of equations, group theory and Galois theory
Scientific career
FieldsMathematics
InfluencesAdrien-Marie Legendre
Joseph-Louis Lagrange
Signature

Contents

Early life

Galois was born on 25 October 1811 to Nicolas-Gabriel Galois and Adélaïde-Marie (née Demante). His father was a Republican and was head of Bourg-la-Reine's liberal party. His father became mayor of the village after Louis XVIII returned to the throne in 1814. His mother, the daughter of a jurist, was a fluent reader of Latin and classical literature and was responsible for her son's education 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, At the age of 14, he began to take a serious interest in mathematics.

He found a copy of Adrien-Marie Legendre's Éléments de Géométrie, which, it is said, he read "like a novel" and mastered at the first reading. At 15, he was reading the original papers of Joseph-Louis Lagrange, such as the Réflexions sur la résolution algébrique des équations which likely motivated his later work on equation theory, and Leçons sur le calcul des fonctions, work intended for professional mathematicians, yet his classwork remained uninspired, and his teachers accused him of affecting ambition and originality in a negative way.

Budding mathematician

In 1828, he attempted the entrance examination for the École Polytechnique, the most prestigious institution for mathematics in France at the time, without the usual preparation in mathematics, and failed for lack of explanations on the oral examination. In that same year, he entered the École Normale (then known as l'École préparatoire), a far inferior institution for mathematical studies at that time, where he found some professors sympathetic to him.[citation needed]

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

In the following year Galois's first paper, on continued fractions, was published. It was at around the same time that he began making fundamental discoveries in the theory of polynomial equations. He submitted two papers on this topic to the Academy of Sciences. Augustin-Louis Cauchy refereed these papers, but refused to accept them for publication for reasons that still remain unclear. However, in spite of many claims to the contrary, it is widely held that Cauchy recognized the importance of Galois's work, and that he merely suggested combining the two papers into one in order to enter it in the competition for the Academy's Grand Prize in Mathematics. Cauchy, an eminent mathematician of the time, though with political views that were at the opposite end from Galois's, considered Galois's work to be a likely winner.

On 28 July 1829, Galois's father died by suicide after a bitter political dispute with the village priest. A couple of days later, Galois made his second and last attempt to enter the Polytechnique, and failed yet again. It is undisputed that Galois was more than qualified; however, accounts differ on why he failed. More plausible accounts state that Galois made too many logical leaps and baffled the incompetent examiner, which enraged Galois. The recent death of his father may have also influenced his behavior.

Having been denied admission to the École polytechnique, Galois took the Baccalaureate examinations in order to enter the École normale. He passed, receiving his degree on 29 December 1829. His examiner in mathematics reported, "This pupil is sometimes obscure in expressing his ideas, but he is intelligent and shows a remarkable spirit of research."

He submitted his memoir on equation theory several times, but it was never published in his lifetime due to various events. Though his first attempt was refused by Cauchy, in February 1830 following Cauchy's suggestion he submitted it to the Academy's secretary Joseph Fourier, to be considered for the Grand Prix of the Academy. Unfortunately, Fourier died soon after, and the memoir was lost. The prize would be awarded that year to Niels Henrik Abel posthumously and also to Carl Gustav Jacob Jacobi. Despite the lost memoir, Galois published three papers that year, one of which laid the foundations for Galois theory. The second one was about the numerical resolution of equations (root finding in modern terminology). The third was an important one in number theory, in which the concept of a finite field was first articulated.

Political firebrand

Battle for the Town Hall by Jean-Victor Schnetz. Galois, as a staunch republican, would have wanted to participate in the July Revolution of 1830 but was prevented by the director of the École Normale.

Galois lived during a time of political turmoil in France. Charles X had succeeded Louis XVIII in 1824, but in 1827 his party suffered a major electoral setback and by 1830 the opposition liberal party became the majority. Charles, faced with political opposition from the chambers, staged a coup d'état, and issued his notorious July Ordinances, touching off the July Revolution which ended with Louis Philippe becoming king. While their counterparts at the Polytechnique were making history in the streets during les Trois Glorieuses, Galois, at the École Normale, was locked in by the school's director. Galois was incensed and wrote a blistering letter criticizing the director, which he submitted to the Gazette des Écoles, signing the letter with his full name. Although the Gazette's editor omitted the signature for publication, Galois was expelled.

Although his expulsion would have formally taken effect on 4 January 1831, Galois quit school immediately and joined the staunchly Republican artillery unit of the National Guard. He divided his time between his mathematical work and his political affiliations. Due to controversy surrounding the unit, soon after Galois became a member, on 31 December 1830, the artillery of the National Guard was disbanded out of fear that they might destabilize the government. At around the same time, nineteen officers of Galois's former unit were arrested and charged with conspiracy to overthrow the government.

In April 1831, the officers were acquitted of all charges, and on 9 May 1831, a banquet was held in their honor, with many illustrious people present, such as Alexandre Dumas. The proceedings grew riotous. At some point, Galois stood and proposed a toast in which he said, "To Louis Philippe," with a dagger above his cup. The republicans at the banquet interpreted Galois's toast as a threat against the king's life and cheered. He was arrested the following day at his mother's house and held in detention at Sainte-Pélagie prison until 15 June 1831, when he had his trial. Galois's defense lawyer cleverly claimed that Galois actually said, "To Louis-Philippe, if he betrays," but that the qualifier was drowned out in the cheers. The prosecutor asked a few more questions, and perhaps influenced by Galois's youth, the jury acquitted him that same day.

On the following Bastille Day (14 July 1831), Galois was at the head of a protest, wearing the uniform of the disbanded artillery, and came heavily armed with several pistols, a loaded rifle, and a dagger. He was again arrested. During his stay in prison, Galois at one point drank alcohol for the first time at the goading of his fellow inmates. One of these inmates, François-Vincent Raspail, recorded what Galois said while drunk in a letter from 25 July. Excerpted from the letter:

And I tell you, I will die in a duel on the occasion of some coquette de bas étage. Why? Because she will invite me to avenge her honor which another has compromised.
Do you know what I lack, my friend? I can confide it only to you: it is someone whom I can love and love only in spirit. I've lost my father and no one has ever replaced him, do you hear me...?

The first line is a haunting prophecy of how Galois would in fact die; the second shows how Galois was profoundly affected by the loss of his father. Raspail continues that Galois, still in a delirium, attempted suicide, and that he would have succeeded if his fellow inmates hadn't forcibly stopped him. Months later, when Galois's trial occurred on 23 October, he was sentenced to six months in prison for illegally wearing a uniform. While in prison, he continued to develop his mathematical ideas. He was released on 29 April 1832.

Final days

Siméon Denis Poisson reviewed Galois's paper on equation theory and declared it "incomprehensible".

Galois returned to mathematics after his expulsion from the École Normale, although he continued to spend time in political activities. After his expulsion became official in January 1831, he attempted to start a private class in advanced algebra which attracted some interest, but this waned, as it seemed that his political activism had priority. Siméon Denis Poisson asked him to submit his work on the theory of equations, which he did on 17 January 1831. Around 4 July 1831, Poisson declared Galois's work "incomprehensible", declaring that "[Galois's] argument is neither sufficiently clear nor sufficiently developed to allow us to judge its rigor"; however, the rejection report ends on an encouraging note: "We would then suggest that the author should publish the whole of his work in order to form a definitive opinion." While Poisson's report was made before Galois's 14 July arrest, it took until October to reach Galois in prison. It is unsurprising, in the light of his character and situation at the time, that Galois reacted violently to the rejection letter, and decided to abandon publishing his papers through the Academy and instead publish them privately through his friend Auguste Chevalier. Apparently, however, Galois did not ignore Poisson's advice, as he began collecting all his mathematical manuscripts while still in prison, and continued polishing his ideas until his release on 29 April 1832, after which he was somehow talked into a duel.

Galois's fatal duel took place on 30 May. The true motives behind the duel are obscure. There has been much speculation as to the reasons behind it. What is known is that, five days before his death, he wrote a letter to Chevalier which clearly alludes to a broken love affair.

Some archival investigation on the original letters suggests that the woman of romantic interest was Stéphanie-Félicie Poterin du Motel, the daughter of the physician at the hostel where Galois stayed during the last months of his life. Fragments of letters from her, copied by Galois himself (with many portions, such as her name, either obliterated or deliberately omitted), are available. The letters hint that du Motel had confided some of her troubles to Galois, and this might have prompted him to provoke the duel himself on her behalf. This conjecture is also supported by other letters Galois later wrote to his friends the night before he died. Galois's cousin, Gabriel Demante, when asked if he knew the cause of the duel, mentioned that Galois "found himself in the presence of a supposed uncle and a supposed fiancé, each of whom provoked the duel." Galois himself exclaimed: "I am the victim of an infamous coquette and her two dupes."

Much more detailed speculation based on these scant historical details has been interpolated by many of Galois's biographers (most notably by Eric Temple Bell in Men of Mathematics), such as the frequently repeated speculation that the entire incident was stage-managed by the police and royalist factions to eliminate a political enemy.

As to his opponent in the duel, Alexandre Dumas names Pescheux d'Herbinville, who was actually one of the nineteen artillery officers whose acquittal was celebrated at the banquet that occasioned Galois's first arrest. However, Dumas is alone in this assertion, and if he were correct it is unclear why d'Herbinville would have been involved. It has been speculated that he was du Motel's "supposed fiancé" at the time (she ultimately married someone else), but no clear evidence has been found supporting this conjecture. On the other hand, extant newspaper clippings from only a few days after the duel give a description of his opponent (identified by the initials "L.D.") that appear to more accurately apply to one of Galois's Republican friends, most probably Ernest Duchatelet, who was imprisoned with Galois on the same charges. Given the conflicting information available, the true identity of his killer may well be lost to history.

Whatever the reasons behind the duel, Galois was so convinced of his impending death that he stayed up all night writing letters to his Republican friends and composing what would become his mathematical testament, the famous letter to Auguste Chevalier outlining his ideas, and three attached manuscripts. Mathematician Hermann Weyl said of this testament, "This letter, if judged by the novelty and profundity of ideas it contains, is perhaps the most substantial piece of writing in the whole literature of mankind." However, the legend of Galois pouring his mathematical thoughts onto paper the night before he died seems to have been exaggerated. In these final papers, he outlined the rough edges of some work he had been doing in analysis and annotated a copy of the manuscript submitted to the Academy and other papers.

The Galois memorial in the cemetery of Bourg-la-Reine. Évariste Galois was buried in a common grave and the exact location is still unknown.

Early in the morning of 30 May 1832, he was shot in the abdomen, was abandoned by his opponents and his own seconds, and was found by a passing farmer. He died the following morning at ten o'clock in the Hôpital Cochin (probably of peritonitis), after refusing the offices of a priest. His funeral ended in riots. There were plans to initiate an uprising during his funeral, but during the same time frame the leaders heard of General Jean Maximilien Lamarque's death, and the rising was postponed without any uprising occurring until 5 June. Only Galois's younger brother was notified of the events prior to Galois's death. He was 20 years old. His last words to his younger brother Alfred were:

"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!)

On 2 June, Évariste Galois was buried in a common grave of the Montparnasse Cemetery whose exact location is unknown. In the cemetery of his native town – Bourg-la-Reine – a cenotaph in his honour was erected beside the graves of his relatives.

In 1843 Joseph Liouville reviewed his manuscript and declared it sound. It was finally published in the October–November 1846 issue of the Journal de Mathématiques Pures et Appliquées. The most famous contribution of this manuscript was a novel proof that there is no quintic formula – that is, that fifth and higher degree equations are not generally solvable by radicals. Although Niels Henrik Abel had already proved the impossibility of a "quintic formula" by radicals in 1824 and Paolo Ruffini had published a solution in 1799 that turned out to be flawed, Galois's methods led to deeper research in what is now called Galois theory. For example, one can use it to determine, for any polynomial equation, whether it has a solution by radicals.

The final page of Galois's mathematical testament, in his own hand. The phrase "to decipher all this mess" ("déchiffrer tout ce gâchis") is on the second to the last line.

From the closing lines of a letter from Galois to his friend Auguste Chevalier, dated 29 May 1832, two days before Galois's death:

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

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

(Ask Jacobi or Gauss publicly to give their opinion, not as to the truth, but as to the importance of these theorems. Later there will be, I hope, some people who will find it to their advantage to decipher all this mess.)

Within the 60 or so pages of Galois's collected works are many important ideas that have had far-reaching consequences for nearly all branches of mathematics. His work has been compared to that of Niels Henrik Abel, another mathematician who died at a very young age, and much of their work had significant overlap.

Algebra

While many mathematicians before Galois gave consideration to what are now known as groups, it was Galois who was the first to use the word group (in French groupe) in a sense close to the technical sense that is understood today, making him among the founders of the branch of algebra known as group theory. He developed the concept that is today known as a normal subgroup. He called the decomposition of a group into its left and right cosets a proper decomposition if the left and right cosets coincide, which is what today is known as a normal subgroup. He also introduced the concept of a finite field (also known as a Galois field in his honor), in essentially the same form as it is understood today.

In his last letter to Chevalier and attached manuscripts, the second of three, he made basic studies of linear groups over finite fields:

Galois theory

Main article: Galois theory

Galois's most significant contribution to mathematics is his development of Galois theory. He realized that the algebraic solution to a polynomial equation is related to the structure of a group of permutations associated with the roots of the polynomial, the Galois group of the polynomial. He found that an equation could be solved in radicals if one can find a series of subgroups of its Galois group, each one normal in its successor with abelian quotient, or its Galois group is solvable. This proved to be a fertile approach, which later mathematicians adapted to many other fields of mathematics besides the theory of equations to which Galois originally applied it.

Analysis

Galois also made some contributions to the theory of Abelian integrals and continued fractions.

As written in his last letter, Galois passed from the study of elliptic functions to consideration of the integrals of the most general algebraic differentials, today called Abelian integrals. He classified these integrals into three categories.

Continued fractions

In his first paper in 1828, Galois proved that the regular continued fraction which represents a quadratic surd ζ is purely periodic if and only if ζ is a reduced surd, that is, ζ > 1 {\displaystyle \zeta >1} and its conjugate η {\displaystyle \eta } satisfies 1 < η < 0 {\displaystyle -1<\eta <0} .

In fact, Galois showed more than this. He also proved that if ζ is a reduced quadratic surd and η is its conjugate, then the continued fractions for ζ and for (−1/η) are both purely periodic, and the repeating block in one of those continued fractions is the mirror image of the repeating block in the other. In symbols we have

ζ = [ a 0 ; a 1 , a 2 , , a m 1 ¯ ] 1 η = [ a m 1 ; a m 2 , a m 3 , , a 0 ¯ ] {\displaystyle {\begin{aligned}\zeta &=[\,{\overline {a_{0};a_{1},a_{2},\dots ,a_{m-1}}}\,]\\[3pt]{\frac {-1}{\eta }}&=[\,{\overline {a_{m-1};a_{m-2},a_{m-3},\dots ,a_{0}}}\,]\,\end{aligned}}}

where ζ is any reduced quadratic surd, and η is its conjugate.

From these two theorems of Galois a result already known to Lagrange can be deduced. If r > 1 is a rational number that is not a perfect square, then

r = [ a 0 ; a 1 , a 2 , , a 2 , a 1 , 2 a 0 ¯ ] . {\displaystyle {\sqrt {r}}=\left[\,a_{0};{\overline {a_{1},a_{2},\dots ,a_{2},a_{1},2a_{0}}}\,\right].}

In particular, if n is any non-square positive integer, the regular continued fraction expansion of √n contains a repeating block of length m, in which the first m − 1 partial denominators form a palindromic string.

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  4. Stewart, Ian (1973).Galois Theory. London: Chapman and Hall. pp. xvii–xxii. ISBN 978-0-412-10800-6.
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  6. "Réflexions sur la résolution algébrique des équations". britannica encyclopedia.
  7. Galois, Évariste (1828). "Démonstration d'un théorème sur les fractions continues périodiques". Annales de Mathématiques. XIX: 294.
  8. Rothman, Tony (1982). "Genius and Biographers: The Fictionalization of Evariste Galois". The American Mathematical Monthly. 89 (2): 84–106. doi:10.2307/2320923. JSTOR 2320923. Retrieved31 January 2015.
  9. 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.
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  11. Galois, Évariste (1830). "Note sur la résolution des équations numériques". Bulletin des Sciences Mathématiques. XIII: 413.
  12. Galois, Évariste (1830). "Sur la théorie des nombres". Bulletin des Sciences Mathématiques. XIII: 428.
  13. Dupuy, Paul (1896). "La vie d'Évariste Galois". Annales Scientifiques de l'École Normale Supérieure. 13: 197–266. doi:10.24033/asens.427.
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  17. Taton, R. (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. doi:10.3406/rhs.1947.2607.
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  19. Infantozzi, Carlos Alberti (1968). "Sur la mort d'Évariste Galois". Revue d'Histoire des Sciences et de Leurs Applications. 21 (2): 157. doi:10.3406/rhs.1968.2554.
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  21. Blanc, Louis (1844). The History of Ten Years, 1830–1840, Volume 1. London: Chapman and Hall. p. 431.
  22. Dalmas, Andre (1956). Évariste Galois: Révolutionnaire et Géomètre. Paris: Fasquelle.
  23. Galois, Évariste (1846). "Lettre de Galois à M. Auguste Chevalier". Journal de Mathématiques Pures et Appliquées. XI: 408–415. Retrieved4 February 2009.
  24. Coutinho, S.C. (1999).The Mathematics of Ciphers. Natick: A K 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. Galois, Évariste (1846). "OEuvres mathématiques d'Évariste Galois". Journal de Mathématiques Pures et Appliquées. XI: 381–444. Retrieved4 February 2009.
  27. 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.
  28. Lie, Sophus (1895). "Influence de Galois sur le Développement des Mathématiques". Le centenaire de l'École Normale 1795–1895. Hachette.
  29. See also: 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.
  30. Letter, p. 410
  31. Letter, p. 411
  32. Wilson, Robert A. (2009). "Chapter 1: Introduction". The finite simple groups. Graduate Texts in Mathematics 251. 251. Berlin, New York: Springer-Verlag. doi:10.1007/978-1-84800-988-2. ISBN 978-1-84800-987-5. Zbl 1203.20012, 2007 preprintCS1 maint: postscript (link)
  33. Letter, pp. 411–412
  34. Galois's last letter, translated
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Évariste Galois
Evariste Galois Language Watch Edit Galois redirects here For other uses see Gallois disambiguation Evariste Galois ɡ ae l ˈ w ɑː 1 French evaʁist ɡalwa 25 October 1811 31 May 1832 was a French mathematician and political activist While still in his teens he was able to determine a necessary and sufficient condition for a polynomial to be solvable by radicals thereby solving a problem that had been open for 350 years His work laid the foundations for Galois theory and group theory 2 two major branches of abstract algebra He was a staunch republican and was heavily involved in the political turmoil that surrounded the French Revolution of 1830 As a result of his political activism he was arrested repeatedly serving one jail sentence of several months For reasons that remain obscure shortly after his release from prison he fought in a duel and died of the wounds he suffered 3 Evariste GaloisA portrait of Evariste Galois aged about 15Born 1811 10 25 25 October 1811 Bourg la Reine French EmpireDied31 May 1832 1832 05 31 aged 20 Paris Kingdom of FranceAlma materEcole preparatoireKnown forWork on theory of equations group theory and Galois theoryScientific careerFieldsMathematicsInfluencesAdrien Marie Legendre Joseph Louis LagrangeSignature Contents 1 Life 1 1 Early life 1 2 Budding mathematician 1 3 Political firebrand 1 4 Final days 2 Contributions to mathematics 2 1 Algebra 2 2 Galois theory 2 3 Analysis 2 4 Continued fractions 3 See also 4 Notes 5 References 6 External linksLife EditEarly life Edit Galois was born on 25 October 1811 to Nicolas Gabriel Galois and Adelaide Marie nee Demante 2 4 His father was a Republican and was head of Bourg la Reine s liberal party His father became mayor of the village 2 after Louis XVIII returned to the throne in 1814 His mother the daughter of a jurist was a fluent reader of Latin and classical literature and was responsible for her son s education for his first twelve years The Cour d honneur of the Lycee Louis le Grand which Galois attended as a boy In October 1823 he entered the Lycee Louis le Grand 5 At the age of 14 he began to take a serious interest in mathematics 5 He found a copy of Adrien Marie Legendre s Elements de Geometrie which it is said he read like a novel and mastered at the first reading At 15 he was reading the original papers of Joseph Louis Lagrange such as the Reflexions sur la resolution algebrique des equations which likely motivated his later work on equation theory 6 and Lecons sur le calcul des fonctions work intended for professional mathematicians yet his classwork remained uninspired and his teachers accused him of affecting ambition and originality in a negative way 4 Budding mathematician Edit In 1828 he attempted the entrance examination for the Ecole Polytechnique the most prestigious institution for mathematics in France at the time without the usual preparation in mathematics and failed for lack of explanations on the oral examination In that same year he entered the Ecole Normale then known as l Ecole preparatoire a far inferior institution for mathematical studies at that time where he found some professors sympathetic to him citation needed Augustin Louis Cauchy reviewed Galois s early mathematical papers In the following year Galois s first paper on continued fractions 7 was published It was at around the same time that he began making fundamental discoveries in the theory of polynomial equations He submitted two papers on this topic to the Academy of Sciences Augustin Louis Cauchy refereed these papers but refused to accept them for publication for reasons that still remain unclear However in spite of many claims to the contrary it is widely held that Cauchy recognized the importance of Galois s work and that he merely suggested combining the two papers into one in order to enter it in the competition for the Academy s Grand Prize in Mathematics Cauchy an eminent mathematician of the time though with political views that were at the opposite end from Galois s considered Galois s work to be a likely winner 8 On 28 July 1829 Galois s father died by suicide after a bitter political dispute with the village priest 9 A couple of days later Galois made his second and last attempt to enter the Polytechnique and failed yet again 9 It is undisputed that Galois was more than qualified however accounts differ on why he failed More plausible accounts state that Galois made too many logical leaps and baffled the incompetent examiner which enraged Galois The recent death of his father may have also influenced his behavior 4 Having been denied admission to the Ecole polytechnique Galois took the Baccalaureate examinations in order to enter the Ecole normale 9 He passed receiving his degree on 29 December 1829 9 His examiner in mathematics reported This pupil is sometimes obscure in expressing his ideas but he is intelligent and shows a remarkable spirit of research He submitted his memoir on equation theory several times but it was never published in his lifetime due to various events Though his first attempt was refused by Cauchy in February 1830 following Cauchy s suggestion he submitted it to the Academy s secretary Joseph Fourier 9 to be considered for the Grand Prix of the Academy Unfortunately Fourier died soon after 9 and the memoir was lost 9 The prize would be awarded that year to Niels Henrik Abel posthumously and also to Carl Gustav Jacob Jacobi Despite the lost memoir Galois published three papers that year one of which laid the foundations for Galois theory 10 The second one was about the numerical resolution of equations root finding in modern terminology 11 The third was an important one in number theory in which the concept of a finite field was first articulated 12 Political firebrand Edit Battle for the Town Hall by Jean Victor Schnetz Galois as a staunch republican would have wanted to participate in the July Revolution of 1830 but was prevented by the director of the Ecole Normale Galois lived during a time of political turmoil in France Charles X had succeeded Louis XVIII in 1824 but in 1827 his party suffered a major electoral setback and by 1830 the opposition liberal party became the majority Charles faced with political opposition from the chambers staged a coup d etat and issued his notorious July Ordinances touching off the July Revolution 9 which ended with Louis Philippe becoming king While their counterparts at the Polytechnique were making history in the streets during les Trois Glorieuses Galois at the Ecole Normale was locked in by the school s director Galois was incensed and wrote a blistering letter criticizing the director which he submitted to the Gazette des Ecoles signing the letter with his full name Although the Gazette s editor omitted the signature for publication Galois was expelled 13 Although his expulsion would have formally taken effect on 4 January 1831 Galois quit school immediately and joined the staunchly Republican artillery unit of the National Guard He divided his time between his mathematical work and his political affiliations Due to controversy surrounding the unit soon after Galois became a member on 31 December 1830 the artillery of the National Guard was disbanded out of fear that they might destabilize the government At around the same time nineteen officers of Galois s former unit were arrested and charged with conspiracy to overthrow the government In April 1831 the officers were acquitted of all charges and on 9 May 1831 a banquet was held in their honor with many illustrious people present such as Alexandre Dumas The proceedings grew riotous At some point Galois stood and proposed a toast in which he said To Louis Philippe with a dagger above his cup The republicans at the banquet interpreted Galois s toast as a threat against the king s life and cheered He was arrested the following day at his mother s house and held in detention at Sainte Pelagie prison until 15 June 1831 when he had his trial 8 Galois s defense lawyer cleverly claimed that Galois actually said To Louis Philippe if he betrays but that the qualifier was drowned out in the cheers The prosecutor asked a few more questions and perhaps influenced by Galois s youth the jury acquitted him that same day 8 9 13 14 On the following Bastille Day 14 July 1831 Galois was at the head of a protest wearing the uniform of the disbanded artillery and came heavily armed with several pistols a loaded rifle and a dagger He was again arrested 9 During his stay in prison Galois at one point drank alcohol for the first time at the goading of his fellow inmates One of these inmates Francois Vincent Raspail recorded what Galois said while drunk in a letter from 25 July Excerpted from the letter 8 And I tell you I will die in a duel on the occasion of some coquette de bas etage Why Because she will invite me to avenge her honor which another has compromised Do you know what I lack my friend I can confide it only to you it is someone whom I can love and love only in spirit I ve lost my father and no one has ever replaced him do you hear me The first line is a haunting prophecy of how Galois would in fact die the second shows how Galois was profoundly affected by the loss of his father Raspail continues that Galois still in a delirium attempted 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 six months in prison for illegally wearing a uniform 9 15 16 While in prison he continued to develop his mathematical ideas He was released on 29 April 1832 Final days Edit Simeon Denis Poisson reviewed Galois s paper on equation theory and declared it incomprehensible Galois returned to mathematics after his expulsion from the Ecole Normale although he continued to spend time in political activities After his expulsion became official in January 1831 he attempted to start a private class in advanced algebra which attracted some interest but this waned as it seemed that his political activism had priority 4 8 Simeon Denis Poisson asked him to submit his work on the theory of equations which he did on 17 January 1831 Around 4 July 1831 Poisson declared Galois s work incomprehensible declaring that Galois s argument is neither sufficiently clear nor sufficiently developed to allow us to judge its rigor however the rejection report ends on an encouraging note We would then suggest that the author should publish the whole of his work in order to form a definitive opinion 17 While Poisson s report was made before Galois s 14 July arrest it took until October to reach Galois in prison It is unsurprising in the light of his character and situation at the time that Galois reacted violently to the rejection letter and decided to abandon publishing his papers through the Academy and instead publish them privately through his friend Auguste Chevalier Apparently however Galois did not ignore Poisson s advice as he began collecting all his mathematical manuscripts while still in prison and continued polishing his ideas until his release on 29 April 1832 13 after which he was somehow talked into a duel 9 Galois s fatal duel took place on 30 May 18 The true motives behind the duel are obscure There has been much speculation as to the reasons behind it What is known is that five days before his death he wrote a letter to Chevalier which clearly alludes to a broken love affair 8 Some archival investigation on the original letters suggests that the woman of romantic interest was Stephanie Felicie Poterin du Motel 19 the daughter of the physician at the hostel where Galois stayed during the last months of his life Fragments of letters from her copied by Galois himself with many portions such as her name either obliterated or deliberately omitted are available 20 The letters hint that du Motel had confided some of her troubles to Galois and this might have prompted him to provoke the duel himself on her behalf This conjecture is also supported by other letters Galois later wrote to his friends the night before he died Galois s cousin Gabriel Demante when asked if he knew the cause of the duel mentioned that Galois found himself in the presence of a supposed uncle and a supposed fiance each of whom provoked the duel Galois himself exclaimed I am the victim of an infamous coquette and her two dupes 13 Much more detailed speculation based on these scant historical details has been interpolated by many of Galois s biographers most notably by Eric Temple Bell in Men of Mathematics such as the frequently repeated speculation that the entire incident was stage managed by the police and royalist factions to eliminate a political enemy 15 As to his opponent in the duel Alexandre Dumas names Pescheux d Herbinville 14 who was actually one of the nineteen artillery officers whose acquittal was celebrated at the banquet that occasioned Galois s first arrest 21 However Dumas is alone in this assertion and if he were correct it is unclear why d Herbinville would have been involved It has been speculated that he was du Motel s supposed fiance at the time she ultimately married someone else but no clear evidence has been found supporting this conjecture On the other hand extant newspaper clippings from only a few days after the duel give a description of his opponent identified by the initials L D that appear to more accurately apply to one of Galois s Republican friends most probably Ernest Duchatelet who was imprisoned with Galois on the same charges 22 Given the conflicting information available the true identity of his killer may well be lost to history Whatever the reasons behind the duel Galois was so convinced of his impending death that he stayed up all night writing letters to his Republican friends and composing what would become his mathematical testament the famous letter to Auguste Chevalier outlining his ideas and three attached manuscripts 23 Mathematician Hermann Weyl said of this testament This letter if judged by the novelty and profundity of ideas it contains is perhaps the most substantial piece of writing in the whole literature of mankind However the legend of Galois pouring his mathematical thoughts onto paper the night before he died seems to have been exaggerated 8 In these final papers he outlined the rough edges of some work he had been doing in analysis and annotated a copy of the manuscript submitted to the Academy and other papers The Galois memorial in the cemetery of Bourg la Reine Evariste Galois was buried in a common grave and the exact location is still unknown Early in the morning of 30 May 1832 he was shot in the abdomen 18 was abandoned by his opponents and his own seconds and was found by a passing farmer He died the following morning 18 at ten o clock in the Hopital Cochin probably of peritonitis after refusing the offices of a priest His funeral ended in riots 18 There were plans to initiate an uprising during his funeral but during the same time frame the leaders heard of General Jean Maximilien Lamarque s death and the rising was postponed without any uprising occurring until 5 June Only Galois s younger brother was notified of the events prior to Galois s death 24 He was 20 years old His last words to his younger brother Alfred were Ne pleure pas Alfred J ai besoin de tout mon courage pour mourir a vingt ans Don t cry Alfred I need all my courage to die at twenty On 2 June Evariste Galois was buried in a common grave of the Montparnasse Cemetery whose exact location is unknown 18 16 In the cemetery of his native town Bourg la Reine a cenotaph in his honour was erected beside the graves of his relatives 25 In 1843 Joseph Liouville reviewed his manuscript and declared it sound It was finally published in the October November 1846 issue of the Journal de Mathematiques Pures et Appliquees 26 27 The most famous contribution of this manuscript was a novel proof that there is no quintic formula that is that fifth and higher degree equations are not generally solvable by radicals Although Niels Henrik Abel had already proved the impossibility of a quintic formula by radicals in 1824 and Paolo Ruffini had published a solution in 1799 that turned out to be flawed Galois s methods led to deeper research in what is now called Galois theory For example one can use it to determine for any polynomial equation whether it has a solution by radicals Contributions to mathematics Edit The final page of Galois s mathematical testament in his own hand The phrase to decipher all this mess dechiffrer tout ce gachis is on the second to the last line From the closing lines of a letter from Galois to his friend Auguste Chevalier dated 29 May 1832 two days before Galois s death 23 Tu prieras publiquement Jacobi ou Gauss de donner leur avis non sur la verite mais sur l importance des theoremes Apres cela il y aura j espere des gens qui trouveront leur profit a dechiffrer tout ce gachis Ask Jacobi or Gauss publicly to give their opinion not as to the truth but as to the importance of these theorems Later there will be I hope some people who will find it to their advantage to decipher all this mess Within the 60 or so pages of Galois s collected works are many important ideas that have had far reaching consequences for nearly all branches of mathematics 28 29 His work has been compared to that of Niels Henrik Abel another mathematician who died at a very young age and much of their work had significant overlap Algebra Edit While many mathematicians before Galois gave consideration to what are now known as groups it was Galois who was the first to use the word group in French groupe in a sense close to the technical sense that is understood today making him among the founders of the branch of algebra known as group theory He developed the concept that is today known as a normal subgroup He called the decomposition of a group into its left and right cosets a proper decomposition if the left and right cosets coincide which is what today is known as a normal subgroup 23 He also introduced the concept of a finite field also known as a Galois field in his honor in essentially the same form as it is understood today 12 In his last letter to Chevalier 23 and attached manuscripts the second of three he made basic studies of linear groups over finite fields He constructed the general linear group over a prime field GL n p and computed its order in studying the Galois group of the general equation of degree pn 30 He constructed the projective special linear group PSL 2 p Galois constructed them as fractional linear transforms and observed that they were simple except if p was 2 or 3 31 These were the second family of finite simple groups after the alternating groups 32 He noted the exceptional fact that PSL 2 p is simple and acts on p points if and only if p is 5 7 or 11 33 34 Galois theory Edit Main article Galois theory Galois s most significant contribution to mathematics is his development of Galois theory He realized that the algebraic solution to a polynomial equation is related to the structure of a group of permutations associated with the roots of the polynomial the Galois group of the polynomial He found that an equation could be solved in radicals if one can find a series of subgroups of its Galois group each one normal in its successor with abelian quotient or its Galois group is solvable This proved to be a fertile approach which later mathematicians adapted to many other fields of mathematics besides the theory of equations to which Galois originally applied it 28 Analysis Edit Galois also made some contributions to the theory of Abelian integrals and continued fractions As written in his last letter 23 Galois passed from the study of elliptic functions to consideration of the integrals of the most general algebraic differentials today called Abelian integrals He classified these integrals into three categories Continued fractions Edit In his first paper in 1828 7 Galois proved that the regular continued fraction which represents a quadratic surd z is purely periodic if and only if z is a reduced surd that is z gt 1 displaystyle zeta gt 1 and its conjugate h displaystyle eta satisfies 1 lt h lt 0 displaystyle 1 lt eta lt 0 In fact Galois showed more than this He also proved that if z is a reduced quadratic surd and h is its conjugate then the continued fractions for z and for 1 h are both purely periodic and the repeating block in one of those continued fractions is the mirror image of the repeating block in the other In symbols we have z a 0 a 1 a 2 a m 1 1 h a m 1 a m 2 a m 3 a 0 displaystyle begin aligned zeta amp overline a 0 a 1 a 2 dots a m 1 3pt frac 1 eta amp overline a m 1 a m 2 a m 3 dots a 0 end aligned where z is any reduced quadratic surd and h is its conjugate From these two theorems of Galois a result already known to Lagrange can be deduced If r gt 1 is a rational number that is not a perfect square then r a 0 a 1 a 2 a 2 a 1 2 a 0 displaystyle sqrt r left a 0 overline a 1 a 2 dots a 2 a 1 2a 0 right In particular if n is any non square positive integer the regular continued fraction expansion of n contains a repeating block of length m in which the first m 1 partial denominators form a palindromic string See also EditGroup theory List of things named after Evariste Galois Niels Henrik AbelNotes Edit Galois theory Random House Webster s Unabridged Dictionary 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 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 a b c d Stewart Ian 1973 Galois Theory London Chapman and Hall pp xvii xxii ISBN 978 0 412 10800 6 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 Reflexions sur la resolution algebrique des equations britannica encyclopedia a b Galois Evariste 1828 Demonstration d un theoreme sur les fractions continues periodiques Annales de Mathematiques XIX 294 a b c d e f g h Rothman Tony 1982 Genius and Biographers The Fictionalization of Evariste Galois The American Mathematical Monthly 89 2 84 106 doi 10 2307 2320923 JSTOR 2320923 Retrieved 31 January 2015 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 Galois Evariste 1830 Analyse d un Memoire sur la resolution algebrique des equations Bulletin des Sciences Mathematiques XIII 271 Galois Evariste 1830 Note sur la resolution des equations numeriques Bulletin des Sciences Mathematiques XIII 413 a b Galois Evariste 1830 Sur la theorie des nombres Bulletin des Sciences Mathematiques XIII 428 a b c d Dupuy Paul 1896 La vie d Evariste Galois Annales Scientifiques de l Ecole Normale Superieure 13 197 266 doi 10 24033 asens 427 a b Dumas pere Alexandre CCIV Mes Memoires ISBN 978 1 4371 5595 2 Retrieved 13 April 2010 a b Bell Eric Temple 1986 Men of Mathematics New York Simon and Schuster ISBN 978 0 671 62818 5 a b Escofier Jean Pierre 2001 Galois Theory Springer pp 222 224 ISBN 978 0 387 98765 1 Taton R 1947 Les relations d Evariste Galois avec les mathematiciens de son temps Revue d Histoire des Sciences et de Leurs Applications 1 2 114 130 doi 10 3406 rhs 1947 2607 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 Infantozzi Carlos Alberti 1968 Sur la mort d Evariste Galois Revue d Histoire des Sciences et de Leurs Applications 21 2 157 doi 10 3406 rhs 1968 2554 Bourgne R J P Azra 1962 Ecrits et memoires mathematiques d Evariste Galois Paris Gauthier Villars Blanc Louis 1844 The History of Ten Years 1830 1840 Volume 1 London Chapman and Hall p 431 Dalmas Andre 1956 Evariste Galois Revolutionnaire et Geometre Paris Fasquelle a b c d e Galois Evariste 1846 Lettre de Galois a M Auguste Chevalier Journal de Mathematiques Pures et Appliquees XI 408 415 Retrieved 4 February 2009 Coutinho S C 1999 The Mathematics of Ciphers Natick A K Peters Ltd pp 127 128 ISBN 978 1 56881 082 9 Toti Rigatelli Laura 1996 Evariste Galois 1811 1832 Vita mathematica 11 Birkhauser p 114 ISBN 978 3 7643 5410 7 Galois Evariste 1846 OEuvres mathematiques d Evariste Galois Journal de Mathematiques Pures et Appliquees XI 381 444 Retrieved 4 February 2009 Pierpont James 1899 Review Oeuvres mathematiques d Evariste Galois publiees sous les auspices de la Societe Mathematique 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 a b Lie Sophus 1895 Influence de Galois sur le Developpement des Mathematiques Le centenaire de l Ecole Normale 1795 1895 Hachette See also Sophus Lie Influence de Galois sur le developpement des mathematiques in Evariste Galois Oeuvres Mathematiques publiees en 1846 dans le Journal de Liouville Sceaux France Editions Jacques Gabay 1989 appendix pages 1 9 Letter p 410 Letter p 411 Wilson Robert A 2009 Chapter 1 Introduction The finite simple groups Graduate Texts in Mathematics 251 251 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 Letter pp 411 412 Galois s last letter translatedReferences EditArtin Emil 1998 Galois Theory Dover Publications Inc ISBN 978 0 486 62342 9 Reprinting of second revised edition of 1944 The University of Notre Dame Press Astruc Alexandre 1994 Evariste Galois Grandes Biographies in French Flammarion ISBN 978 2 08 066675 8 Bell E T 1937 Galois Men of Mathematics 2 Still in print Deserable Francois Henri 2015 Evariste in French Gallimard ISBN 9782070147045 Edwards Harold M May 1984 Galois Theory Graduate Texts in Mathematics 101 Springer Verlag ISBN 978 0 387 90980 6 This textbook explains Galois Theory with historical development and includes an English translation of Galois s memoir Ehrhardt Caroline 2011 Evariste Galois la fabrication d une icone mathematique En temps et lieux in French Editions de l Ecole Pratiques de Hautes Etudes en Sciences Sociales ISBN 978 2 7132 2317 4 Infeld Leopold 1948 Whom the Gods Love The Story of Evariste Galois Classics in Mathematics Education Series Reston Va National Council of Teachers of Mathematics ISBN 978 0 87353 125 2 Classic fictionalized biography by physicist Infeld Livio Mario 2006 The Equation That Couldn t Be Solved How Mathematical Genius Discovered the Language of Symmetry Physics Today Souvenir Press 59 7 50 Bibcode 2006PhT 59g 50L doi 10 1063 1 2337831 ISBN 978 0 285 63743 6 Toti Rigatelli Laura 1996 Evariste Galois Birkhauser ISBN 978 3 7643 5410 7 This biography challenges the common myth concerning Galois s duel and death Stewart Ian 1973 Galois Theory Chapman and Hall ISBN 978 0 412 10800 6 This comprehensive text on Galois Theory includes a brief biography of Galois himself Tignol Jean Pierre 2001 Galois theory of algebraic equations Singapore World Scientific ISBN 978 981 02 4541 2 Historical development of Galois theory External links EditWikimedia Commons has media related to Evariste Galois Wikiquote has quotations related to Evariste GaloisWikisource has the text of the 1911 Encyclopaedia Britannica article Galois Evariste Works by Evariste Galois at Project Gutenberg Works by or about Evariste Galois at Internet Archive O Connor John J Robertson Edmund F Evariste Galois MacTutor History of Mathematics archive University of St Andrews The Galois Archive biography letters and texts in various languages Two Galois articles online and analyzed on BibNum Memoire sur les conditions de resolubilite des equations par radicaux 1830 link for English analysis click A telecharger Demonstration d un theoreme sur les fractions continues periodiques 1829 link for English analysis click A telecharger Rothman Tony 1982 Genius and Biographers The Fictionalization of Evariste Galois PDF The American Mathematical Monthly 89 2 84 106 doi 10 2307 2320923 JSTOR 2320923 La vie d Evariste Galois by Paul Dupuy The first and still one of the most extensive biographies referred to by every other serious biographer of Galois Œuvres Mathematiques published in 1846 in the Journal de Liouville converted to Djvu format by Prof Antoine Chambert Loir at the University of Rennes Alexandre Dumas Mes Memoires the relevant chapter of Alexandre Dumas memoires where he mentions Galois and the banquet Evariste Galois at the Mathematics Genealogy Project Theatrical trailer of University College Utrecht s Evariste En Garde A piece of music dedicated to Evariste Galois on YouTube Retrieved from https en wikipedia org w index php title Evariste Galois amp oldid 1051853813, wikipedia, wiki, book,

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