Abel was born on August 5, 1802 in the village of Finnøy, in the diocese of Kristiansand. Both his father and several of his ancestors had distinguished themselves as ecclesiastics, and all of his paternal ancestors were educated. Abel’s mother, Anna-Maria Simonsen, was known for her beauty and Abel inherited her appearance. At the age of thirteen, he enrolled with his older brother at the Catedral de Cristianía School, a very demanding school, where Abel’s genius was able to manifest itself precociously.
During that period, the school did not have good teachers, the best, in fact, had been called to the University of Cristianía (now Oslo), open to teaching since 1813. When a very strict school teacher was fired by his brutal methods that had caused the death of a child, he was replaced by a not very brilliant but very valuable mathematician, Bernt Michael Holmboe (1795-1850), who was of great help to Abel. Under his leadership, Abel began to fully assimilate many works of great mathematicians, particularly the works of Newton, Euler, Joseph-Louis Lagrange, and Gauss’s Disquisitiones Aritmeticae. Since then, mathematics became Niels Abel’s main occupation, and also his greatest fun. Abel, with his critical sense, who was one of the first to discover gaps in the reasoning of his predecessors and from a young age he was eager to fill them.
One of his works in this line of thought is the first demonstration of the development of the binomial series general, integrated into the broader context of the theory and application of the infinite series.
Abel’s father died in 1820, so the responsibility of the family fell on his shoulders and, to meet the family needs, he began to give private lessons, dedicating the few free moments to his mathematical research. He entered the University of Cristianía in 1821 with mediocre grades in all subjects except mathematics. Convinced that he had before his eyes one of the greatest mathematicians of all time, Holmboe did everything possible to obtain subsidies for him; In those same years Abel contracted the virus of the disease that would lead him prematurely to death.
His first job was the search for a solution to the general fifth degree equation of the polynomial, problem not yet solved by his predecessors: he was able to show that no general algebraic solution of this equation was possible.
He then set out to study these two interrelated problems:
– find all the algebraic equations of different degrees that could be solved algebraically;
– determine whether or not a given algebraic equation can be solved algebraically.
However, it was another young mathematical genius of the time, Galois, who provided the necessary and sufficient conditions for an algebraic equation to be solved algebraically.
State funding allowed him to visit Germany and France in 1825. Abel met the astronomer Schumacher (1780-1850) in Altona near Hamburg, when he spent six months in Berlin, where he worked on the beginning of the publication of the mathematical journal of August Leopold Crelle. . This project was strongly supported by Abel, who contributed a lot to the success of the initiative. From Berlin he went to Freiberg, where he conducted his brilliant research in the theory of functions, especially taking into account ellipticals and hyperelipics, and the introduction of a new class of functions that are now known as abelian functions.
In 1826, he traveled to Paris, where he stayed for about ten months, there he met the greatest French mathematicians, however, both he and his little-known work were little appreciated, also because of his modesty that did not lead him to declare the results of their studies. During his stay in Paris, Abel consulted the doctors for what he considers to be a persistent cold, the doctors diagnosed him with pulmonary tuberculosis, but not believing in this diagnosis, he traveled to Berlin, where he stayed for two months.
Financial difficulties, from which he had never escaped, forced him to interrupt his journey to return to Norway, where he taught for some time in Cristiania. In early April 1829 Crelle helped him get a job at the University of Berlin, but the offer reached Norway two days after his death from pneumonia.
The premature death of this talented mathematician ended a brilliant and promising career. His research has cleared some of the greatest darkness in mathematical analysis, opening up new fields of study and allowing many ramifications and considerable advances in mathematical knowledge.
Abel’s deepest and most original work was published in Crelle’s Journal of which Holmboe was editor. The most complete edition of his work was published in 1881 by Ludwig Sylow and Sophus Lie. The abelian adjective, common in mathematical writings, derives from its name and is conventionally represented by a lowercase letter.
In 2002, the prestigious Abel Prize, created to fill the void of the Nobel Prize, which continues to ignore mathematics, was established in his honor. Abel is also the name of a lunar crater, which recalls the name of the famous Norwegian mathematician; it is located in the southern-eastern hemisphere facing the Earth. It is located south of Barnard crater, in the northeastern part of the Southern Ocean. The relief that defines the perimeter of the crater has an irregular and highly distorted shape, and shows the traces of numerous impacts.
He is also credited with the statement of the theorem known as Abel-Ruffini, which postulates that general polynomial equations of degree equal to or greater than five cannot be solved by radicals.
The theorem was first proved by the Italian Paolo Ruffini in 1799, but his proof was generally ignored; Abel studied this problem and published a proof in 1824.