Neils Henrik Abel Father of Group theory

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Neils Henrik Abel Father of Group theory
« on: April 23, 2013, 05:01:24 PM »
Niels Henrik Abel (1802–1829) Norwegian Algebra Born on August 5, 1802.


Niels Abel might have been one of the great mathematicians of the 19th century had he not died of tuberculosis at age 26. He is remembered, and honored, in mathematics for putting an end to the three-century-long search for a solution by radicals of the quintic equation. His theoretical work in the topics of group theory and algebra paved the way for continued significant research in these areas.
          Abel’s short life was dominated by poverty, chiefly due to the severe economic hardships his homeland of Norway endured after the Napoleonic wars, exacerbated by difficult family circumstances. A schoolteacher, thankfully, recognized Abel’s talent for mathematics as a young student and introduced him to the works of Leonhard Euler, Joseph-Louis Lagrange, and other great mathematicians. He also helped raise money to have Abel attend university and continue his studies. Abel entered the University of Christiania in the city of Christiania (present-day Oslo), Norway, in 1821.
          During his final year of study, Abel began working on the solution of quintic equations (fifth-degree polynomial equations) by radicals. Although scholars for a long time knew general formulae for solving for quadratic, cubic, and quartic equations using nothing more than basic arithmetical operations on the coefficients that appear in the equation, no one had yet found a similar formula for solving quintics. In 1822 Abel believed he had produced one. He shared the details of his method with the Danish mathematician Ferdinand Degen in hopes of having the work published by the Royal Society of Copenhagen. Degen had trouble following the logic behind Abel’s approach and asked for a numerical illustration of his method. While trying to produce a numerical example, Abel found an error in his paper that eventually led him to understand the reason why general solutions to fifth- and higher-degree equations are impossible. Abel published this phenomenal discovery in 1825 in a self-published pamphlet “Mémoire sur les équations algébriques où on démontre l’impossibilité de la résolution de l’équation générale ducinquième degré” (Memoir on the impossibility of algebraic solutions to the general equations of the fifth degree), which he later presented as a series of seven papers in the newly established Journal for Pure and Applied Mathematics (commonly known as Crelle’s Journal for its German founder August Leopold Crelle). At first, reaction to this work was slow, but as the reputation of the journal grew, more and more scholars took note of the paper, and news of Abel’s accomplishment began to spread across Europe. A few years later Abel was honored with a professorship at the University of Berlin. Unfortunately, Abel had contracted tuberculosis by this time, and he died on April 6, 1829, a few days before receiving the letter of notification.
          In 1830 the Paris Academy awarded Abel, posthumously, the Grand Prix for his outstanding work. Although Abel did not write in terms of the modern-day concepts of group theory, mathematicians call groups satisfying the commutative property “Abelian groups” in his honor. In 2002, on the bicentenary of his birth, the Norwegian Academy of Science and Letters created a new mathematics prize, the Abel Prize, similar to the Nobel Prize, to be awarded annually.
          Research in the field of commutative algebra continues today using the approach developed by Abel during his short life. His influence on the development of Abstract Algebra is truly significant.


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