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These two men, though contemporaries, are not related by nationality or similar mathematical interest; each, like a streaking meteor in the mathematical heavens, flashed to an early brilliance and then was suddenly and pathetically extinguished by premature death, leaving remarkable material for future mathematicians to work upon. Abel died of tuberculosis and malnutrition in his twenty-sixth year, and Galois died in a foolish duel in his twenty-first year; neither man was properly appreciated during his lifetime for his genius. |
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Abel, born in 1820 at Findo in Norway, was the son of a country minister.
nbsp; When a student in
Christiania, he
&thought he had discovered how to solve the
general quintic equation algebraically, but soon corrected himself in a famous
pamphlet published in 1824. In this early paper, Abel established the impossiblility of
solving the general quintic equation by means of radicals, thus finally
laying to rest a difficult problem that had puzzled mathematicians from Bombelli to Viete.
As a result of this paper.. Abel obtained a small
stipend that permitted him to travel in germany, ltaly, and France. During
these travels, He wrote a number of papers in various areas of mathematics
such as on the convergence of infinite series on the so-called Abelian integrals,
and on elliptic funcions. Abel's researches on elliptic functions arose in exciting and friendly competiton with Jacobi. The older Legendre, who had done pioneer work on elliptic functions, was deeply impressed with Abel's discoveries. Luckily Abel secured an outlet for his papers in the newly founded journal fur die reine und angewandte Mathematik (more popularly known as Crelle's journal): in fact, the first volume of the journal(1826) contained no less than five of Abel's papers, and the second volume(1827) contained Abel's work that gave birth to the theory of doubly periodic functions. Every student of analysis encounters Abel's integral equation and Abel's theorem on the sum of integrals of algebraic functions that leads to Abelian functions. In infinite series work there is Abel's convergence test and Abel's theorem on power series. In abstract algebra, commutative groups are today called Abelian groups. Plagued by poverty all his life and suffering from a pulmonary condition. Abel was unable to obtain a teaching position. He died tragically at Froland in. Norway in 1829. Two days after his death, a delayed letter was delivered in which Abel was belatedly offered a teaching post at the University of Berlin. Although Abel received little recognition from his government when alive, he now appears on some of his country's smaller-denominational postage stamps. But the mathematicians in their characteristic manner have erected far more lasting monuments to Abel for today Abel's name is perpetuated in an abundance of theorems and theories. Of Abel Hermite once said, "He has left mathematicians something to keep them busy for five hundred years." When asked his formula for so rapidly forging ahead to the first ranks of his discipline, Abel replied, "By studying the masters and not their pupils." | |
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Evariste Galois had an even shorter and more tragic life than did Abel. Born near Paris in 1811 as the son of a small-town mayor, he began to exhibit an extraordinary mathematical talent shortly after his fifteenth birthday. He tried twice to enter the Ecole Polytechnique But both times was refused admission because of his inability to meet the formal requirements of his examiners, who completely failed to recognize his genius. Then came another blow;his father, feeling himself persecuted by the clerics, committed suicide. |
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Persevering, Galois
finally entered the Ecole
Normale in 1829 to prepare himself to teach, but,
drawn by democratic sympathy into the turmoils of the
Revolution of 1830, hewas expelled
from school and spent several months in prison. Shortly after his
release, in 1832 when not yet twenty-one years old, he was manipulated into a
pistol duel over a love affair and was slain. Galois mastered the mathematical textbooks of his time with the ease of reading novels, went on to the important papers of Legendre, Jacobi and Abel, and then turned to creating mathematics of his own. In his seventeenth year he reached results of great importance but two memoirs that he sent to the French Academy were mislaid and lost adding to his frustration. A short paper of his on equations was published in 1830 and gave results apparently based on a very general theory. The night before his duel, fully realizing he would in all probability be killed, he wrote his scientific testament in the form of a letter to one of his friends. This testament referred to some of his unpublished discoveries, the later unraveling of which required the talents of some great mathematicians, and that turned out to contain the theory of groups and the so-called Galois theory of equations. The Galois theory of equations, based upon concepts of group theory, supplies criteria for the possibility of solving an algebraic equation by radicals. Galois essentially created the study of groups;he was the first(in1830) to use the word "group" in its technical sense. Researches in the theory of groups were then carried on by Augustin-Louis Cauchy (1789-1857) and his succes sors under the particular guise of substitution groups.
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