Niels Henrik Abel was born in Nedstrand, Norway, as the second child of Søren Georg Abel and Anne Marie Simonsen. When he was born, the family was living at a rectory on Finnøy. Much suggests that Niels Henrik was born in the neighboring parish, as his parents were guests of the bailiff in Nedstrand in July / August of his year of birth.  Niels Henrik Abel’s father, Søren Georg Abel, had a degree in theology and philosophy and served as pastor at Finnøy. Søren’s father, Niels’s grandfather, Hans Mathias Abel, was also a pastor, at Gjerstad near Risør. Søren had spent his childhood at Gjerstad, and had also served as chaplain there; and after his father’s death in 1804, Søren was appointed pastor at Gjerstad and the family moved there.

Anne Marie Simonsen was from Risør; her father, Niels Henrik Saxild Simonsen, was a tradesman and merchant ship-owner, and said to be the richest person in Risør. Anne Marie had grown up with two stepmothers, in relatively luxurious surroundings. At Gjerstad rectory, she enjoyed arranging balls and social gatherings. Much suggests she was early on an alcoholic and took little interest in the upbringing of the children. Niels Henrik and his brothers were given their schooling by their father, with handwritten books to read. Interestingly, an addition table in a book of mathematics reads: 1+0=0.

With Norwegian independence and the first election held in Norway, in 1814, Søren Abel was elected as a representative to the Storting. Meetings of the Storting were held until 1866 in the main hall of the Cathedral School in Christiania (now known as Oslo). Almost certainly, this is how he came into contact with the school, and he decided that his eldest son, Hans Mathias, should start there the following year. However, when the time for his departure approached, Hans was so saddened and depressed over having to leave home that his father did not dare send him away. He decided to send Niels instead.

In 1815, Niels Abel entered the Cathedral School at the age of 13. His elder brother Hans joined him there a year later. They shared rooms and had classes together. Hans got better grades than Niels; however, a new mathematics teacher, Bernt Michael Holmboe, was appointed in 1818. He gave the students mathematical tasks to do at home. He saw Niels Henrik’s talent in mathematics, and encouraged him to study the subject to an advanced level. He even gave Niels private lessons after school.

In 1818, Søren Abel had a public theological argument with the theologian Stener Johannes Stenersen regarding his catechism from 1806. The argument was well covered in press. Søren was given the nickname “Abel Treating” (Norwegian: “Abel Spandabel”). Niels’ reaction to the quarrel was said to have been “excessive gaiety”. At the same time, Søren also almost faced impeachment after insulting Carsten Anker, the host of the Norwegian Constituent Assembly; and in September 1818 he returned to Gjerstad with his political career in ruins. He began drinking heavily and died only two years later, in 1820, aged 48.  Bernt Michael Holmboe supported Niels Henrik Abel with a scholarship to remain at the school and raised money from his friends to enable him to study at the Royal Frederick University.

When Abel entered the university in 1821, he was already the most knowledgeable mathematician in Norway. Holmboe had nothing more he could teach him and Abel had studied all the latest mathematical literature in the university library. During this time Abel started working on the quintic equation in radicals. Abel initially thought he had found the solution to the quintic equation in radicals in 1821. Mathematicians had been looking for a solution on this problem for over 250 years. The two professors in Christiania, Søren Rasmussen and Christopher Hansteen, found no errors in Abel’s formulas, and sent the work on to the leading mathematician in the Nordic countries, Professor Ferdinand Degen in Copenhagen. He also found no faults, but still doubted that the solution, which so many outstanding mathematicians had sought for so long, could now really have been found by an unknown student in far-off Christiania. Degen noted, however, Abel’s unusually sharp mind, and believed that such a talented young man should not waste his abilities on such a “sterile object” as the fifth degree equation, but rather on elliptic functions and transcendence; for then, writes Degen, he will “discover Magellanian thoroughfares to large portions of a vast analytical ocean”. Degen asked Abel to give a numerical example of his method and, while trying to provide an example, Abel discovered a mistake in his paper.  Abel graduated in 1822. His performance was average, except in mathematics which was exceptionally high.

After he graduated, professors from university supported Abel financially, and Professor Christopher Hansteen let him live in a room in the attic of his home. Abel would later view Ms. Hansteen as his second mother. While living here, Abel helped his younger brother, Peder Abel, through examen artium. He also helped his sister Elisabeth to find work in the town.

In early 1823, Niels Abel published his first article in “Magazin for Naturvidenskaberne”, Norway’s first scientific journal, which had been co-founded by Professor Hansteen. Abel published several articles, but the journal soon realized that this was not material for the common reader. In 1823, Abel also wrote a paper in French. It was “a general representation of the possibility to integrate all differential formulas” (Norwegian: en alminnelig Fremstilling af Muligheten at integrere alle mulige Differential-Formler). He applied for funds at the university to publish it. However the work was lost, while being reviewed, never to be found thereafter.

In mid-1823, Professor Rasmussen gave Abel a gift of 100 speciedaler so he could travel to Copenhagen and visit Ferdinand Degen and other mathematicians there. While in Copenhagen, Abel did some work on Fermat’s Last Theorem. Abel’s uncle, Peder Mandrup Tuxen, lived at the naval base in Christianshavn, Copenhagen, and at a ball there Niels Abel met Christine Kemp, his future fiancée. In 1824, Christine moved to Son, Norway to work as a governess and the couple got engaged over Christmas.

After returning from Copenhagen, Abel applied for a government scholarship in order to visit top mathematicians in Germany and France, but he was instead granted 200 speciedaler yearly for two years, to stay in Cristiania and study German and French. In the next two years, he was promised a scholarship of 600 speciedaler yearly and he would then be permitted to travel abroad. While studying these languages, Abel published his first notable work in 1824, Mémoire sur les équations algébriques où on démontre l’impossibilité de la résolution de l’équation générale du cinquième degré (Memoir on algebraic equations, in which the impossibility of solving the general equation of the fifth degree is proven). For, in 1823, Abel had at last proved the impossibility of solving the quintic equation in radicals (now referred to as the Abel–Ruffini theorem). However, this paper was in an abstruse and difficult form, in part because he had restricted himself to only six pages, in order to save money on printing. A more detailed proof was published in 1826 in the first volume of Crelle’s Journal.

In 1825, Abel wrote a personal letter to King Carl Johan of Norway/Sweden requesting permission to travel abroad. He was granted this permission, and in September 1825 he left Christiania together with four friends from university (Christian P.B Boeck, Balthazar M. Keilhau, Nicolay B. Møller and Otto Tank). These four friends of Abel were traveling to Berlin and to the Alps to study geology. Abel wanted to follow them to Copenhagen and from there make his way to Göttingen. The terms for his scholarship stipulated that he was to visit Gauss in Göttingen and then continue to Paris. However, when he got as far as Copenhagen he changed his plans. He wanted to follow his friends to Berlin instead, intending to visit Göttingen and Paris afterwards.

On the way, he visited the astronomer Heinrich Christian Schumacher in Altona, now a district of Hamburg. He then spent four months in Berlin, where he became well acquainted with August Leopold Crelle, who was then about to publish his mathematical journal, Journal für die reine und angewandte Mathematik. This project was warmly encouraged by Abel, who contributed much to the success of the venture. Abel contributed seven articles to it in its first year.  From Berlin Abel also followed his friends to the Alps. He went to Leipzig and Freiberg to visit Georg Amadeus Carl Friedrich Naumann and his brother the mathematician August Naumann. In Freiberg Abel did brilliant research in the theory of functions, particularly, elliptic, hyperelliptic, and a new class now known as abelian functions.

From Freiberg they went on to Dresden, Prague, Vienna, Trieste, Venice, Verona, Bolzano, Innsbruck, Luzern and Basel. From July 1826 Abel traveled on his own from Basel to Paris. Abel had sent most of his work to Berlin to be published in Crelles Journal, but he had saved what he regarded as his most important work for the French Academy of Sciences, a theorem on addition of algebraic differentials. With the help of Johan Gørbitz he found an apartment in Paris and continued his work on the theorem. He finished in October 1826, and submitted it to the academy. It was to be reviewed by Augustin-Louis Cauchy. Abel’s work was scarcely known in Paris, and his modesty restrained him from proclaiming his research. The theorem was put aside and forgotten until his death.

Abel’s limited finances finally compelled him to abandon his tour in January 1827. He returned to Berlin, and was offered a position as editor of Crelles Journal, but opted out. By May 1827 he was back in Norway. His tour abroad was viewed as a failure. He had not visited Gauss in Göttingen and he had not published anything in Paris. His scholarship was therefore not renewed and he had to take up a private loan in Norges Bank of 200 spesidaler. He never repaid this loan. He also started tutoring. He continued to send most of his work to Crelles Journal. But in mid-1828 he published, in rivalry with Carl Jacobi, an important work on elliptic functions in Astronomische Nachrichten in Altona.

Abel showed that there is no general algebraic solution for the roots of a quintic equation, or any general polynomial equation of degree greater than four, in terms of explicit algebraic operations. To do this, he invented (independently of Galois) an extremely important branch of mathematics known as group theory, which is invaluable not only in many areas of mathematics, but for much of physics as well. Abel sent a paper on the unsolvability of the quintic equation to Carl Friedrich Gauss, who proceeded to discard without a glance what he believed to be the worthless work of a crank. Before Abel solved the problem, it had gone unsolved for over 250 years. He originally thought he had found a solution in 1821, which turned out to be wrong.

As a 16 year old Abel gave a proof of the binomial theorem valid for all numbers, extending Euler’s result which had held only for rationals. Abel wrote a fundamental work on the theory of elliptic integrals, containing the foundations of the theory of elliptic functions. While travelling to Paris he published a paper revealing the double periodicity of elliptic functions, which Adrien-Marie Legendre later described to Augustin-Louis Cauchy as “a monument more lasting than bronze” (borrowing a famous sentence by the Roman poet Horatius). The paper was however, misplaced by Cauchy.

While abroad Abel had sent most of his work to Berlin to be published in the Crelles Journal, but he had saved what he regarded as his most important work for the French Academy of Sciences, a theorem on addition of algebraic differentials. The theorem was put aside and forgotten until his death. While in Freiberg Abel did brilliant research in the theory of functions, particularly, elliptic, hyperelliptic, and a new class now known as abelian functions.

In 1823 Abel wrote a paper titled “a general representation of the possibility to integrate all differential formulas” (Norwegian: en alminnelig Fremstilling af Muligheten at integrere alle mulige Differential-Formler). He applied for funds at the university to publish it. However the work was lost, while being reviewed, never to be found thereafter.  Abel said famously of Carl Friedrich Gauss’s writing style, “He is like the fox, who effaces his tracks in the sand with his tail.”

While in Paris, Abel contracted tuberculosis. During the Christmas of 1828, he traveled by sled to Froland to visit his fiancée. He became seriously ill on the journey and, although a temporary improvement allowed the couple to enjoy the holiday together, he died relatively soon after on 6 April 1829, just two days before a letter arrived from August Crelle. Crelle had been searching for a new job for Abel in Berlin, and had actually managed to have him appointed as a professor at the University of Berlin. Crelle wrote to Abel on 8 April 1829 to tell him the good news, but it came too late.