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.

In October 1823, he entered the Lycée Louis-le-Grand, At the age of 14, he began to take a serious interest in mathematics.

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.

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.

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} .

On 28 July 1829, Galois’ father committed 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.

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’ 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’ 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’ 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’ youth, the jury acquitted him that same day.

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’ 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. 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.

The first line is a haunting prophecy of how Galois will 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’ 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.

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’ 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.