Editing Évariste Galois (section)

=== Final days ===
[[File:Simeon Poisson.jpg|thumb|right|[[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.<ref name="stewart-gt" /><ref name="rothman" /> [[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."<ref>{{cite journal
 | last = Taton
 | first = R.
 | year = 1947
 | title = Les relations d'Évariste Galois avec les mathématiciens de son temps
 | journal = Revue d'Histoire des Sciences et de Leurs Applications
 | volume = 1
 | issue = 2
 | pages = 114–130
 | doi = 10.3406/rhs.1947.2607| url = http://www.persee.fr/doc/rhs_0048-7996_1947_num_1_2_2607
 }}</ref> 
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&nbsp;April 1832,<ref name="dupuy" /> after which he was somehow talked into a duel.<ref name=":2" />

Galois's fatal duel took place on 30 May.<ref name=":3">{{Cite book|title=Math and mathematicians : the history of math discoveries around the world|last=C.|first=Bruno, Leonard|orig-year=1999|year=2003|publisher=U X L|others=Baker, Lawrence W.|isbn=978-0787638139|location=Detroit, Mich.|page=[https://archive.org/details/mathmathematicia00brun/page/174 174]|oclc=41497065|url=https://archive.org/details/mathmathematicia00brun|url-access=registration}}</ref> The true motives behind the duel are obscure. There has been much speculation about them. What is known is that, five days before his death, he wrote a letter to Chevalier which clearly alludes to a broken love affair.<ref name="rothman" />

Some archival investigation on the original letters suggests that the woman of romantic interest was Stéphanie-Félicie Poterin du Motel,<ref>{{cite journal
 | last = Infantozzi
 | first = Carlos Alberti
 | year = 1968
 | title = Sur la mort d'Évariste Galois
 | journal = Revue d'Histoire des Sciences et de Leurs Applications
 | volume = 21
 | issue = 2
 | page = 157
 | doi = 10.3406/rhs.1968.2554
}}</ref> 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.<ref>{{cite book
  | last = Bourgne
  | first = R.
  |author2=J.-P. Azra
  | title = Écrits et mémoires mathématiques d'Évariste Galois
  | publisher = Gauthier-Villars
  | location = Paris
  | year = 1962}}</ref> The letters hint that Poterin 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."<ref name="dupuy" /> 

As to his opponent in the duel, Alexandre Dumas names Pescheux d'Herbinville,<ref name="dumas"/> who was actually one of the nineteen artillery officers whose acquittal was celebrated at the banquet that occasioned Galois's first arrest.<ref name="blanc">
{{cite book
  | last = Blanc
  | first = Louis
  | title = The History of Ten Years, 1830–1840, Volume 1
  | publisher = Chapman and Hall
  | location = London
  | year = 1844
  | page = [https://archive.org/details/bub_gb_hH52mPMmlzcC/page/n441 431]
  | url = https://archive.org/details/bub_gb_hH52mPMmlzcC
}}
</ref> 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 Poterin 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.<ref name="dalmas">
{{cite book
  | last = Dalmas
  | first = Andre
  | title = Évariste Galois: Révolutionnaire et Géomètre
  | publisher = Fasquelle
  | location = Paris
  | year = 1956 }}
</ref> 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.<ref name="chevalier-letter">{{Cite web |title=Galois' last letter in nLab |url=https://ncatlab.org/nlab/show/Galois'+last+letter |access-date=2025-05-15 |website=ncatlab.org}}</ref> 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.<ref name="rothman" /> 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.

[[File:Tombe galois.JPG|thumb|left|The Galois memorial in the cemetery of [[Bourg-la-Reine]]. Évariste Galois was buried in a common grave and the exact location is unknown.]]
Early in the morning of 30 May 1832, he was shot in the [[abdomen]],<ref name=":3" /> was abandoned by his opponents and his own seconds, and was found by a passing farmer. He died the following morning<ref name=":3" /> at ten o'clock in the [[Hôpital Cochin]] (probably of [[peritonitis]]), after refusing the offices of a priest. His funeral ended in riots.<ref name=":3" /> There were plans to initiate an uprising during his funeral, but during the same time the leaders heard of General [[Jean Maximilien Lamarque]]'s death and the rising was postponed without any uprising occurring until [[June Rebellion|5 June]]. Only Galois's younger brother was notified of the events prior to Galois's death.<ref>{{cite book|last=Coutinho|first=S.C.|title=The Mathematics of Ciphers|url=https://archive.org/details/mathematicsofcip0000cout|url-access=registration|year=1999|publisher=A K Peters, Ltd.|location=Natick|isbn=978-1-56881-082-9|pages=[https://archive.org/details/mathematicsofcip0000cout/page/127 127–128]}}</ref> Galois was 20 years old. His [[last words]] to his younger brother Alfred were:
{{blockquote|"Ne pleure pas, Alfred&nbsp;! J'ai besoin de tout mon courage pour mourir à vingt ans&nbsp;!"<br />(Don't weep, 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.<ref name=":3"/><ref name="escofier">{{cite book
| last = Escofier
  | first = Jean-Pierre
  | title = Galois Theory
  | url = https://archive.org/details/springer_10.1007-978-1-4613-0191-2
  | publisher = Springer
  | year = 2001
  | pages = [https://archive.org/details/springer_10.1007-978-1-4613-0191-2/page/n228 222]–224
  | isbn = 978-0-387-98765-1}}</ref> In the cemetery of his native town – [[Bourg-la-Reine]] – a [[cenotaph]] in his honour was erected beside the graves of his relatives.<ref>{{cite book
  | last = Toti Rigatelli
  | first = Laura |author-link = Laura Toti Rigatelli
  | title = Evariste Galois, 1811–1832 (Vita mathematica, 11)
  | publisher = Birkhäuser
  | year = 1996
  | page = [https://archive.org/details/evaristegalois180000toti/page/114 114]
  | isbn = 978-3-7643-5410-7
  | url = https://archive.org/details/evaristegalois180000toti/page/114
  }}</ref>

Évariste Galois died in 1832. Joseph Liouville began studying Galois's unpublished papers in 1842 and acknowledged their value in 1843. It is not clear what happened in the 10 years between 1832 and 1842 nor what eventually inspired Joseph Liouville to begin reading Galois's papers. Jesper Lützen explores this subject at some length in Chapter XIV ''Galois Theory'' of his book about [[Joseph Liouville]] without reaching any definitive conclusions.<ref name="Lützen">{{cite book
  | last = Lützen
  | first = Jesper
  | title = Joseph Liouville 1809–1882: Master of Pure and Applied Mathematics|series = Studies in the History of Mathematics and Physical Sciences
  | year = 1990
  | chapter = Chapter XIV: Galois Theory
  | pages = 559–580
  | volume = 15
  | publisher = Springer-Verlag
  | isbn = 3-540-97180-7
  }}</ref>

It is certainly possible that mathematicians (including Liouville) did not want to publicize Galois's papers because Galois was a republican political activist who died 5 days before the [[June Rebellion]], an unsuccessful anti-monarchist insurrection of Parisian republicans. In Galois's obituary, his friend Auguste Chevalier almost accused academicians at the École Polytechnique of having killed Galois since, if they had not rejected his work, he would have become a mathematician and would not have devoted himself to the republican political activism for which some believed he was killed. Given that France was still living in the shadow of the [[Reign of Terror]] and the [[Napoleonic era]], Liouville might have waited until the political turmoil subsided (from the failed [[June Rebellion]]  and its aftermath) before turning his attention to Galois's papers.<ref name="Lützen" />

Liouville finally published Galois's manuscripts in the October–November 1846 issue of the ''[[Journal de Mathématiques Pures et Appliquées]]''.<ref>{{cite journal
 | last = Galois
 | first = Évariste
 | year = 1846
 | title = OEuvres mathématiques d'Évariste Galois.
 | journal = Journal de Mathématiques Pures et Appliquées
 | volume = XI
 | pages = 381–444
 | url = http://visualiseur.bnf.fr/ark:/12148/cb343487840/date1846
 | access-date = 4 February 2009}}</ref><ref>{{cite journal|author=Pierpont, James|author-link=James Pierpont (mathematician)|title= 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.|journal=Bull. Amer. Math. Soc.|year=1899|volume=5|issue=6|pages=296–300|url=http://www.ams.org/journals/bull/1899-05-06/S0002-9904-1899-00599-8/S0002-9904-1899-00599-8.pdf|doi=10.1090/S0002-9904-1899-00599-8|doi-access=free}} In 1897 the French Mathematical Society reprinted the 1846 publication.</ref> Galois's most famous contribution was a novel proof that there is no [[Quintic equation|quintic formula]] – that is, that fifth and higher degree equations are not generally solvable by radicals. Although [[Niels Henrik Abel]] had already [[Abel–Ruffini theorem|proved the impossibility of a "quintic formula" by radicals]] in 1824 and [[Paolo Ruffini (mathematician)|Paolo Ruffini]] had published a solution in 1799 that turned out to be flawed, Galois's methods led to deeper research into what is now called [[Galois Theory]], which  can be used to determine, for ''any'' polynomial equation, whether it has a solution by radicals.