209 years ago, a famous mathematician Évariste Galois was born. He lived only 20 years but managed to become a classic of mathematics and one of the founders of higher algebra. His short biography is well known because it is romantic and interesting to young people. Galois was a fiery revolutionary and romantic who died in a duel (Betti 21). In the last letter to his friend, which Évariste wrote before his death, he outlined the latest discoveries in mathematics. Évariste Galois stands out among the famous scientists of the past since he has managed to make a great discovery in the field of algebra in such a short life.
The famous mathematician was born on October 25, 1811, in the city of Bourg-la-Reine, near Paris. His father Nicolas-Gabriel Galois was a supporter of Napoleon; he was elected mayor of Bourg-la-Reine in 1815, during the Hundred Days War (Nowlan 327). Évariste’s formal education started in 1823, and his teacher Jean Hippolyte Verron-Vernier awakened Évariste’s interest in mathematics (Nowlan 327). Having easily mastered the curriculum, he enthusiastically studied the book of geometer Adrien-Marie Legendre and the works of famous mathematician Joseph-Louis Lagrange.
Galois decided to take the entrance examination to the Polytechnic Institute without a usual preparatory course but failed. However, he continued to make progress in mathematics and enrolled at the Lyceum in a higher-level mathematics class, led by an experienced teacher Louis Paul Émile Richard (Nowlan 328). He realized how gifted Galois was and helped him to enter the Polytechnic Institute without exams. In March 1829, when Galois was still a student, his first article called “A Proof of a Theorem on Periodic Continued Fractions” was published (Nowlan 328). He also turned to the theory of equations, which he studied from the works of Lagrange.
The mathematicians of his time were faced with the task of finding algorithms for solving complex algebraic equations. They did not succeed, and as a result, scientists agreed among themselves that there was no comprehensive formula, and the issue had no solution. The central question was about the method for solving an equation with one unknown x, all the coefficients of which are rational numbers, and the term of the highest degree is equal to xn (Ferreirós and Reck 70). The method should apply to all similar equations and include only four elementary operations and extracting a root. If the solutions can be received from the coefficients of the equation only using these operations, then it is solvable by radicals.
Évariste’s accumulated experience suggested that solving an equation of the nth degree would not require more complicated operations than extracting the root of the nth degree. The solution of an equation of the second degree ax2 + bx + c = 0 requires extracting the square root from some combination of coefficients, namely from the expression b2 – 4ac (Sinha 112). In the same way, the general solution of the cubic equation is reduced to extracting the cubic root from a certain combination of coefficients (Nowlan 329). The solution to a general fourth-degree equation, first obtained by the Italian mathematician Lodovico Ferrari at about the same time, requires the extraction of fourth-degree roots.
Before Galois, for almost three hundred years, no one had succeeded in solving the general equation of the fifth degree or higher in radicals. Many mathematicians were inclined to think that a general solution of this kind is impossible. Although, in special cases, for example, in the case of the equation x7 – 2 = 0, the solution can be found in radicals (Sinha 107). Galois found the final criteria that made it possible to determine whether a solution to this equation exists in radicals. His research led to a theory now called group theory, whose applications go far beyond the theory of equations.
Galois presented his first article in the area that would later become group theory to the French Academy of Sciences. It happened on May 25, 1829, shortly before graduating from the Lyceum (Gray 117). Less than two months later, he had to take the entrance examination to the Polytechnic Institute, but events took an unfortunate turn. On July 2, a few weeks before the exam, Évariste’s father committed suicide, unable to endure the scandal around his name (Gray 117). The environment for the exam was unfavorable, and, as a result, Galois failed again.
Évariste was accepted to École Normale thanks to a high score in mathematics. At the same time, he submitted his paper on group theory. In 1830, he went far ahead in the study of conditions determining the solvability of equations, although he had not yet received a complete solution to this problem (Gray 116). In January 1831, he completed the work and submitted it to the Academy of Sciences (Gray 117). This paper was his most significant work, which gave mathematicians food for thought for hundreds of years.
When Galois was finishing his work on group theory, political events burst into his life. In July 1830, Republicans who opposed the restored monarchy took to the streets (Nowlan 328). In the months following the revolution, Évariste attended meetings of the Republicans, met with their leaders, and took part in the unrest and demonstrations that fevered Paris. He joined the National Guard artillery, a militia unit composed almost exclusively of Republicans. After that, he was expelled from the institute and was arrested twice.
However, even after being imprisoned, he continued to conduct mathematical research and did not abandon it until his death. The fact that he could work productively in such conditions speaks of the extraordinary power of his imagination and intelligence. In mid-March 1832, due to the cholera epidemic in Paris, Évariste was transferred from Sainte-Pélagie prison to a private hospital (Nowlan 327). Apparently, there he met the girl, because of whom he died in a duel on May 31, 1832 (Nowlan 328). It was Stephanie-Felice Poterin du Motel, the daughter of a Parisian doctor.
Fulfilling the wish of Évariste, his younger brother Alfred and Auguste Chevalier sent copies of the manuscript to Carl Gauss, Carl Jacobi, and other famous mathematicians. However, it took almost ten years before his work was appreciated. This happened in 1846 when one of the copies was presented to the eminent French mathematician Joseph Liouville (Gray 135). The scientist devoted a lot of time to the work of Galois, edited his memoirs, and published it in his prestigious edition Journal de Mathèmatiques Pures et Appliquées.
Évariste Galois’s mathematical works, at least those that have survived, are only sixty pages long. Never before have works of such a small volume brought the author such wide popularity. Group theory, which this young mathematician pioneered, is now a prolific area of mathematics. Nobody could have guessed that questions about the solvability of equations would lead to one of the key concepts in mathematics or that groups would be the method in which symmetry is described.
Works Cited
Betti, Renato. “Cinema Scientists: A Film about Galois.” Lettera Matematica, vol. 6, no. 1, 2018, pp. 19-23. SpringerLink, Web.
Ferreirós, José, and Erich H. Reck. “Dedekind’s Mathematical Structuralism: From Galois Theory to Numbers, Sets, and Functions.” The Prehistory of Mathematical Structuralism, edited by Erich H. Reck and Georg Schiemer, Oxford UP, 2020, pp. 59-87.
Gray, Jeremy J. A History of Abstract Algebra: From Algebraic Equations to Modern Algebra. Springer, 2018.
Nowlan, Robert A. Masters of Mathematics: The Problems They Solved, Why These Are Important, and What You Should Know about Them. Springer, 2017.
Sinha, Rajnikant. Galois Theory and Advanced Linear Algebra. Springer, 2020.