Niels Henrik Abel

Biography

Niels Henrik Abel was born on 5 August 1802 in Finnøy, Norway, as the second of six children in a poor rural family. His father, Søren Georg Abel, was a Lutheran minister and amateur politician, and the family moved often because of his work. Despite their limited means, Abel showed an exceptional talent for mathematics from an early age. His teachers at Oslo Cathedral School noticed his abilities, particularly Bernt Michael Holmboe, who recognized his genius and gave him advanced mathematical texts and personal tutoring. Holmboe became one of Abel's key early supporters, helping to secure funding for his further education.

Abel joined the University of Oslo and continued to teach himself mathematics passionately, delving into the works of Euler, Lagrange, and Gauss. While still a student, he tackled one of the most famous unsolved problems in mathematics: finding a general formula for solving polynomial equations of the fifth degree, known as the quintic equation. Mathematicians had long sought such a formula, having already solved equations of degree two, three, and four. In 1823, Abel provided the first solid proof that no such algebraic solution in radicals could exist for the general quintic, a discovery that changed the course of algebra.

Despite the importance of his work, Abel spent much of his short career fighting poverty and lack of recognition. He traveled to Berlin and Paris trying to gain acknowledgment from the European mathematical community, carrying with him papers that would later be seen as some of the most important of the nineteenth century. In Paris, he submitted a paper on elliptic functions to the French Academy of Sciences, but Augustin-Louis Cauchy misplaced it, and it remained unpublished until after Abel's death. The French Academy eventually awarded him the Grand prix des sciences mathématiques in 1830, posthumously, for this work.

During his travels and correspondence, Abel made significant strides in the theory of elliptic functions, working alongside and in competition with German mathematician Carl Gustav Jacob Jacobi. He discovered what are now called Abelian functions and established Abel's theorem, which broadly connects integrals of algebraic functions. He also contributed to the theory of infinite series, developing criteria for convergence, including results related to what is now called Abel's theorem on power series.

Abel returned to Norway without a steady academic position, surviving on temporary tutoring jobs and a small stipend. His health, already weakened by harsh travel conditions and poor living situations, worsened quickly. He was diagnosed with tuberculosis and died on 6 April 1829 in Froland, Norway, at the age of 26. Two days after his death, a letter arrived from Berlin offering him a professorship. The French mathematician Charles Hermite later remarked that Abel had left mathematicians enough material to work on for five hundred years, highlighting the impact of his less than seven years of active mathematical work.