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' methods led to deeper research in what is now called Galois theory.

Évariste Galois (; ; 25 October 1811 – 31 May 1832) was a French mathematician and political activist.

He died at age 20 from wounds suffered in a duel. == Life == === Early life === Galois was born on 25 October 1811 to Nicolas-Gabriel Galois and Adélaïde-Marie (née Demante).

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. 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.

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.

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' methods led to deeper research in what is now called Galois theory.

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.

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, \zeta > 1 and its conjugate \eta satisfies -1 . In fact, Galois showed more than this.

Cauchy, an eminent mathematician of the time, though with political views that were at the opposite end from Galois', considered Galois' work to be a likely winner. On 28 July 1829, Galois' father died by suicide after a bitter political dispute with the village priest.

He passed, receiving his degree on 29 December 1829.

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.

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.

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.

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.

At around the same time, nineteen officers of Galois' 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.

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.

The prosecutor asked a few more questions, and perhaps influenced by Galois' 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.

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' work "incomprehensible", declaring that "[Galois'] 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' July 14 arrest, it took until October to reach Galois in prison.

Évariste Galois (; ; 25 October 1811 – 31 May 1832) was a French mathematician and political activist.

He was released on 29 April 1832. === Final days === Galois returned to mathematics after his expulsion from the École Normale, although he continued to spend time in political activities.

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' fatal duel took place on 30 May.

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. 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.

For example, one can use it to determine, for any polynomial equation, whether it has a solution by radicals. == Contributions to mathematics == From the closing lines of a letter from Galois to his friend Auguste Chevalier, dated May 29, 1832, two days before Galois' death: Within the 60 or so pages of Galois' collected works are many important ideas that have had far-reaching consequences for nearly all branches of mathematics.

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.

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