Galois

Evariste Galois

Evariste Galois was a French mathematician who made significant contribution to the theory of functions, the theory of equations and number theory. His work became the basis for Galois Theory and Group theory. He was the first to use the word Group as a technical term in mathematics to represent a group of permutations. 

 The mathematical genius Evariste Galois was born on 25^th October 1811 in France. He was the second son of Nicholas Gabriel Galois and Adelaide Marie. His father was the director of a boarding school and later mayor of Bourg – la- Reive. Evariste had a happy childhood. He seemed to have a phenomenal memory.  While still in teens, he was able to determine a necessary and sufficient condition for a polynomial to be solvable by radicals, thereby solving a long standing problem. Up to the age of 12, he was educated by his mother who instilled in him knowledge of classics and a sceptical attitude towards religion.

He entered the College Royale de Louis le Grand in Paris in 1823 where his precocious mathematical genius first emerged. He studied the Legendre’s text book Elements de Geometrie and Lagrange’s work on Differential Equation and Analytic Function. By the age of 16, he had published many papers. He appeared for the examination of Ecole Polytechnic but failed. The next year Galois had the good fortune to be studying maths under a distinguished teacher Louis Paul Emile Richard who recognised his exceptional gifts. Richard was enthusiastic about Galois mathematical work and this led to his publication of first paper on continued fraction. 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 but the Academy refused to accept them for publication.

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. 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."

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 and Louis Philippe became the king. His anti monarchist view led him to imprisonment leading his expulsion from the college.

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. They were later acquitted of all charges.

During his imprisonment, he continued developing his mathematical ideas. On the advice of Simon Poisson he submitted his work on the Theory of Equation for publication but Poisson decleared his work incomprehensible. Galois reacted violently to the rejection and decided to abandon publishing his papers through the Academy and instead publish them privately through his friend Auguste Chevalier. 

During 1830s Galois published several papers in Bulletin des Sciences mathematiques of Baren de Ferussace, a review that normally only published work by established scientist. These articles contained most of his work on Theory of Equation which is now known as Galois Theory.

In 1831, he was arrested again for keeping illegal weapon. He was released on April 29, 1832. On May 25 he wrote to Chevalier, a close friend of his from the Ecole Normale, expressing his complete disenchantment with life, and hinting that a broken love affair was the reason. The woman in question was Stephanie Dumotel, the daughter of the resident physician at the hostel where Galois stayed during last months of his life.

On May 30, Galois fought a duel with pistol and was shot in the abdomen. Galois remained unattended for hours until a passerby took him to the hospital. He refused the services of a priest and died of peritonitis the following day, at the age of 20. A night before the duel was scheduled he recorded his mathematical idea in a later to his former school master Auguste Chevalier where he had out lined his work on elliptical integrals and permutations of groups.

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."

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.

 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 Abel, another mathematician who died at a very young age, and much of their work had significant overlap.

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