• Mathematician Évariste Galois, 1811
• Mathematician Paolo Ruffini, 1765

Galois was born on 25 October 1811 to Nicolas - Gabriel Galois and Adélaïde - Marie (born Demante). His father was a Republican and was head of Bourg - la - Reine's liberal party. He 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. At the age of 10, Galois was offered a place at the college of Reims, but his mother preferred to keep him at home.

In October 1823, he entered the Lycée Louis - le - Grand, and despite some turmoil in the school at the beginning of the term (when about a hundred students were expelled), Galois managed to perform well for the first two years, obtaining the first prize in Latin. He soon became bored with his studies and, at the age of 14, began to take a serious interest in mathematics.

He found a copy of Adrien Marie Legendre's Éléments de Géométrie, which it is said that 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 landmark 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.

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 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 appears that Cauchy recognized the importance of Galois' 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, considered Galois' work to be a likely winner.

On 28 July 1829 Galois's 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. The legend holds that he thought the exercise proposed to him by the examiner to be of no interest, and, in exasperation, threw at the examiner's head the rag used to erase the blackboard. More plausible accounts state that Galois made too many logical leaps and was enraged by the bafflement of the examiner. The recent death of his father may have also influenced his behavior.

Having been denied admission to the 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. As noted before, his first attempt was refused by Cauchy, but in February 1830 following Cauchy's suggestion he submitted it to the Academy's secretary 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 Abel posthumously and also to 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.

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 abdication, 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 Polytechnique were making history in the streets during les Trois Glorieuses, Galois and all the other students at the École Normale were 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' former unit were arrested and charged with conspiracy to overthrow the government.

In April, 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, and Galois proposed a toast to King Louis - Philippe with a dagger above his cup, which was interpreted as a threat against the king's life. He was arrested the following day, but was acquitted on 15 June.

On the following Bastille Day, Galois was at the head of a protest, wearing the uniform of the disbanded artillery, and came heavily armed with several pistols, a rifle, and a dagger. For this, he was again arrested and this time was sentenced to six months in prison for illegally wearing a uniform. He was released on 29 April 1832. During his imprisonment, he continued developing his mathematical ideas.

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. Simeon Poisson asked him to submit his work on the theory of equations, which he did on 17 January. Around 4 July, 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' Bastille Day 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, began collecting all his mathematical manuscripts while still in prison, and continued polishing his ideas until his release on 29 April 1832.

Galois' fatal duel took place on 30 May. The true motives behind the duel will most likely remain forever obscure. There has been much speculation, much of it spurious, 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 a Mademoiselle 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 either obliterated, such as her name, or deliberately omitted) are available. The letters hint that Mlle. 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. Much more detailed speculation based on these scant historical details has been interpolated by many of Galois' 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, one of the nineteen artillery officers whose acquittal was celebrated at the banquet that occasioned Galois' first arrest and du Motel's fiancé. However, Dumas is alone in this assertion, and extant newspaper clippings from only a few days after the duel give a description of his opponent that more accurately applies to one of Galois' 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. Hermann Weyl, a mathematician, 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.

Early in the morning of 30 May 1832 he was shot in the abdomen and died the following morning at ten o'clock in the Cochin hospital (probably of peritonitis) after refusing the offices of a priest. He was 20 years old. His last words to his 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 honor was erected beside the graves of his relatives.

Galois' mathematical contributions were published in full in 1843 when 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 solvable by radicals. Although Abel had already proved the impossibility of a "quintic formula" by radicals in 1824 and 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.

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.)

Unsurprisingly, Galois' collected works amount to only some 60 pages, but within them 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.

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:

• He constructed the general linear group over a prime field, GL(ν, p) and computed its order, in studying the Galois group of the general equation of degree p^ν.
• 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. These were the second family of finite simple groups, after the alternating groups.
• 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.

Paolo Ruffini (September 22, 1765 – May 10, 1822) was an Italian mathematician and philosopher.

By 1788 he had earned university degrees in philosophy, medicine / surgery, and mathematics. Among his work was an incomplete proof (Abel – Ruffini theorem) that quintic (and higher order) equations cannot be solved by radicals (1799), and Ruffini's rule which is a quick method for polynomial division. Ruffini also made contributions to group theory in addition to probability and quadrature of the circle.

He practiced as both a professor of mathematics (University of Modena) and a doctor including scientific work on typhus.

Ruffini’s 1799 work marked a major development for group theory. Ruffini developed Joseph Louis Lagrange's work on permutation theory, following 29 years after Lagrange’s "Réflexions sur la théorie algébrique des equations" (1770 – 1771) which was largely ignored until Ruffini who established strong connections between permutations and the solvability of algebraic equations. Ruffini was the first to controversially assert the unsolvability by radicals of algebraic equations higher than quartics. This angered many members of the community such as Malfatti (1731 – 1807). Work in this area was later carried on by those such as Gauss and Galois who succeeded in such a proof.

1799: "Teoria Generale delle Equazioni, in cui si dimostra impossibile la soluzione algebraica delle equazioni generali di grado superiore al quarto"

1802: "Riflessioni intorno alla rettificazione ed alla quadratura del circulo"

1802: "Della soluzione delle equazioni algebraiche determinate particolari di grado superiore al quarto"

1804: "Sopra la determinazione delle radici nelle equazioni numeriche di qualunque grado"

1806: "Della immortalità dell’anima"

1807: "Algebra elementare"

1820: "Memoria sul tifo contagioso"

1821: "Riflessioni critiche sopra il saggio filosofico intorno alle probabilità del signor conte Laplace"