Niels Henrik Abel (1802-1829) was a Norwegian mathematician who proved that fifth and higher order equations have no algebraic solution. Had he not died prematurely, it is speculated that he might have become one of the most prominent mathematicians of the 19th century. He provided the first general proof of the binomial theorem and made significant discoveries concerning elliptic functions

# Niels Abel

Abel was born in Finnöy, on the southwestern coast of Norway, on August 5, 1802. He was the second son of Sören Georg Abel, a Lutheran minister, and Anne Marie nee Sorensen, the daughter of a wealthy merchant. Abel's father was appointed to a new parish in 1804, and the family moved to the town of Gjerstad, in southern Norway. Abel received his early education from his father. In 1815, he was sent to the Cathedral School in Oslo, where he soon developed a passion for mathematics. In 1818, a new instructor, Berndt Holmboe, arrived at the school and fueled Abel's interest further, introducing him to the works of such European masters as Isaac Newton, Joseph-Louis Lagrange, and Leonhard Euler. Holmboe was to become a lifelong friend and advocate, eventually helping to raise money that allowed Abel to travel abroad and meet the leading mathematicians of Germany and France.

Abel graduated from the Cathedral School in 1821. His father had died a year earlier and his older brother had developed mental illness. The responsibility of providing for his mother and four younger siblings fell largely on Abel. To make ends meet, he began tutoring. Meanwhile, he took the entrance examination for the university. His performance in geometry and arithmetic was distinguished and he was offered a free dormitory room. In an exceptional move, members of the mathematics faculty, who were already aware of Abel's promise, contributed personal funds to cover his other expenses. Abel enrolled at the University of Kristiania (Oslo) at the age of 19. Within a year he had completed his basic courses and was a degree candidate.

## Proved Impossibility of Solutions for Quintic Problem

During his final year at the Cathedral School, Abel had become intrigued by a challenge that had occupied some of the best mathematical minds since the 16th century, that of finding a solution to the "quintic" problem. A quintic equation is one in which the unknown appears to the fifth power. Abel believed he had discovered a general solution and presented his results to his teacher Holmboe, who was wise enough to realize that the mathematical reasoning of Abel was beyond his full comprehension. Holmboe sent the solution to the Danish mathematician Ferdinand Degen, who expressed skepticism but was unable to determine whether Abel's argument was flawed. Degen asked Abel to provide examples of his general solution, and was eventually able to discover the error in his approach. Abel would remain obsessed with the quintic problem for the next few years. Finally, in 1823, he hit upon the realization and derived a proof that an algebraic solution was impossible. Abel sent a paper describing his proof to Johann Karl Friedrich Gauss, who reportedly ignored the treatise. Meanwhile, Abel began working on what would become the first proof of an integral equation, and went on to provide the first general proof of the binomial theorem, which until then had only been proved for special cases. He also investigated elliptic integrals and developed a novel way of examining them through the use of inverse functions.

In 1825, Abel left home and traveled to Berlin, where he met August Leopold Crelle, a civil engineer and the builder of the first German railroad. Crelle had a strong reverence for mathematics, and was about to publish the first edition of *Journal for Pure and Applied Mathematics,* the first periodical devoted entirely to mathematical research. Recognizing in Abel a man of genius, Crelle asked if the young man would contribute to the premiere edition. Abel obliged, providing Crelle with a manuscript that described his proof that an algebraic solution to the general equation of the fifth and higher degrees was impossible. The paper would insure both Abel's fame and the success of Crelle's fledgling journal. From Germany, Abel toured southern Europe. He then traveled to France, where he made the acquaintance of Adrien Marie Legendre, Augustin Louis Cauchy, and others. In their company, he wrote the *Memoir on a General Property of a Very Extensive Class of Transcendental Functions,* which was submitted to the Paris Académie Royale des Sciences. The memoir expounded on Abel's earlier work on elliptical functions, and proposed what has come to be known as Abel's theorem. Unfortunately, it was received poorly, rejected by Legendre because it was "illegible," then temporarily lost by Cauchy. Two years after Abel's death, the manuscript finally resurfaced, but it was not published until 1841.

By 1827, Abel had run out of money and was forced to return to Norway. He had hoped to take up a university post, but could only find work as a tutor. At this time, he discovered that he had contracted tuberculosis. Later in 1827, he wrote a lengthy paper on elliptic functions for Crelle's journal and began working for Crelle as an editor.

Abel died on April 6, 1829, while visiting his Danish fiancée, Christine Kemp, who was living in Froland. A few days later, unaware of Abel's death, Crelle wrote to say he had secured a position for him at the University of Berlin. Abel was honored posthumously, in 1830, when the French Académie awarded him the Grand Prix, a prize he shared with Karl Jacobi.

## Further Reading on Niels Abel

Bell, E.T., *Men of Mathematics,* Simon and Schuster, 1986. Ore, Oystein, *Niels Henrik Abel: Mathematician Extraordinary,*

University of Minnesota Press, 1957.

"Niels Henrik Abel," *MacTutor History of Mathematics Archive.*
http://www-groups.dcs.st-and.ac.uk/~history/Mathematics/abel.html (March 1997).