Biography of Évariste Galois (1811-1832)

Évariste Galois commemorative stamp.

French mathematician from which came the theory of groups, born 25 October 1811 in Bourg-la-Reine, near Paris. Son of Nicholas-Gabriel Galois supporter of Napoleon and one of the leaders of the liberal party of the city, and of which he/she was Mayor during the 100 days in which Napoleon returned from his exile.

Until the age of twelve was educated by his mother, Adelaide-Marie Demante Galois, who provided him with training in the classical languages, latin, Greek, his skepticism by the Church and basic notions of arithmetic.

In 1823 he/she entered at the College Royal de Louis-le-Grand, in Paris, where he/she developed his ideas of liberal and antiroyalist, moreover already inculcated by his parents. During the first two courses is mainly in the subjects of latin and Greek, which received several awards, but then had to repeat the third grade by its difficulties in rhetoric. At age 15, received their first math course, taught by Hippolyte Jean Vernier, arousing his interest in them; from there begins to read the works of Adrien Marie Legendre, "Elements of Geometrie" and Joseph Louis Lagrange "Resolution of algebraic equations", "The theory of analytic functions", "Lessons on calculation of functions". This passion for mathematics made him forget about the rest of the materials.

In 1827, it occurs to the entrance examination of the École Polytechnique, one of the most prestigious teaching institutions of the period, without prior preparation course required, being rejected.

He then enrolled in a higher mathematics course at the Louis-le-Grand, given by Louis Paul Émile Richard, who observed the endowments of his student applies for its admission into the École Polytechnique without prior examination, request that was rejected. In March 1829, he/she published his first work, "Demonstration of a theorem about periodic continued fractions" in "Annales de mathématiques pures et apliquées" and his attention had already set in the theory of equations.

Send articles on a fledgling theory of groups to the French Academy of Sciences during his last year at the Louis-le-Grand, although without much success. On July 2, 1829, her father commits suicide and the next week should be examined again to enter the École Polytechnique, where is definitely suspended. The created situation, decides to enter the École Normale, where he/she was admitted after passing an examination with excellent qualifications.

The absence of care by the Academy, Galois begins to publish in the Bulletin des sciences mathématiques, astronomiques, physiques et chimiques de Baron de Férussac. There articles show that it had gone further than any other mathematician regarding the conditions that define the solubility of equations, but his theory was not yet complete.

In January 1831, at the request of the mathematician Siméon Denis Poisson, the Academy sends another memory, perhaps the most important work of Galois on the theory of groups, even though Poisson recommends to the Academy that it reject it finally and Galois who develop the work and explain it more clearly, because their arguments are too concise.

Galois is politically, involves contacting and entering Republican societies and participating in political demonstrations of the era. Because of this, and a letter that sends to the director of Ecole Normale in which accuses him of traitor is expelled from it.

In 1832, he/she is imprisoned for a month in the prison of Sainte-Pélagie for providing by Luis Felipe de Orléans, however the jury acquitted him. But shortly thereafter, on July 14, 1831, he/she returns to be arrested for wearing the uniform of the guard of artillery, body which had been dissolved by considering it a threat to the throne; for this reason, Galois spends eight months back in Sainte-Pélagie, where suffers all kinds of calamities, and is even said to try suicide.

Once released, he/she is challenged to a duel of honor by Pescheux d'Herbinville, a political activist who on May 30, 1832, in the course of the duel on the outskirts of Paris wounds him in the abdomen of a bullet; Galois is abandoned to their fate and ends up being picked up by a bystander that he/she transferred to the hospital Cochin, where he/she died shortly afterwards as a result of a peritonitis when he/she had not even met the twenty-one years of age, being buried in the mass grave in the cemetery South.

The night before the celebration of the duel, Galois writes a long letter to Auguste Chevalier, which corrects the drafting of three manuscripts, one work rejected by Poisson, another a condensed version of an article appeared in le Bulletin de Férussac and a third concerning the General algebraic functions integrals; at the same time it asks Chevalier request its opinion about the importance of these theorems C. G. Jacobi and C. F. Gauss. The manuscript of this letter you can read on the phrase "I don't have time," writes Galois perhaps sensing his imminent death.

These manuscripts were published in 1848 by the French mathematician Joseph Liouville, thus being born the mathematical branch of the theory of groups.

Scientific work

At the time of Galois, the problem of the theory of equations arose under the prism of find a general method of resolution of equations, polynomial with a single question and order n-th, based on the four elementary operations of arithmetic (addition, subtraction, multiplication and division) and the extraction of roots.

The importance of the work of Galois was not so much the criteria that sat down to determine if a polynomial equation solutions might be found by radical or not, but that idea for this study and methods from which was born the theory of groups, one application that is much greater than the theory of equations.

The group theory deals with intrinsic Symmetries of one system either, and Galois introduces three fundamental notions, that allow you to demonstrate that no general method for equations of degree higher than the fourth, allowing its resolution using the operations of addition, subtraction, multiplication, division and extraction of roots there is:

1. each equation is associated with a group of permutations; This group is a representation of the symmetry properties of the equation that is now called Galois group.

2. notion of normal subgroup; a subgroup of a group is normal it is verified that by multiplying by the left any element of the subgroup by an element either parental group g and then multiplying by right the resulting product by the inverse of g element, the result is still an element of the subgroup.

3. notion of soluble group; a group is soluble when it generates a maximal normal subgroups whose composition factors are all cousins; such factors are determined from the numbers of elements of the parental group and subgroups.

Each equation of degree n is associated with a group known as Galois group, or some subset of this; Galois showed that they were only resolvable by arithmetic procedures and extraction of roots, those equations whose Galois group is soluble; and he/she showed that effectively had equal or superior to the fifth degree equations for which the corresponding group is not soluble.