Эварист Галуа: Жизнь и математические открытия

French mathematician (1811–1832)

Évariste Galois (/g//æ//l/'/w//ɑː/; evaʁist ɡalwa; 25 October 1811 – 31 May 1832) was a French mathematician and political activist. While still in his teens, he was able to determine a necessary and sufficient condition for a polynomial to be solvable by radicals, thereby solving a problem that had been open for 350 years. His work laid the foundations for Galois theory and group theory, two major branches of abstract algebra. Galois was a staunch republican and was heavily involved in the political turmoil that surrounded the French Revolution of 1830. As a result of his political activism, he was arrested repeatedly, serving one jail sentence of several months. For reasons that remain obscure, shortly after his release from prison, Galois fought in a duel and died of the wounds he suffered.

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 At the age of 14, he began to take a serious interest in mathematics. and Leçons sur le calcul des fonctions, work intended for professional mathematicians, yet his classwork remained uninspired and his teachers accused him of putting on the airs of a genius. 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 is widely held that Cauchy recognized the importance of Galois's 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 though with political views that were diametrically opposed to those of Galois, considered Galois's work to be a likely winner. On 28 July 1829, Galois's father died by 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. The second 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.

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 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 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.

Galois's most significant contribution to mathematics is his development of Galois theory. He realized that the algebraic solution to a polynomial equation is related to the structure of a group of permutations associated with the roots of the polynomial, the Galois group of the polynomial. He found that an equation could be solved in radicals if one can find a series of subgroups of its Galois group, each one normal in its successor with abelian quotient, that is, its Galois group is solvable. This proved to be a fertile approach, which later mathematicians adapted to many other fields of mathematics besides the theory of equations to which Galois originally applied it.

Galois also made some contributions to the theory of Abelian integrals and continued fractions. As written in his last letter, Galois passed from the study of elliptic functions to consideration of the integrals of the most general algebraic differentials, today called Abelian integrals. He classified these integrals into three categories.