Niels Henrik Abel was born on August 5, 1802 in Frindoe, Norway. His father was a vicar in the village of Gjerstad. His father was active politically and was part of the movement that led to Norway's separation from Denmark in 1814.
When Niels was 13 years old, he left home to attend the Cathedral School in Christiania. The school was set up in the new ideals of the time. Instead of corporal punishment, in theory, teachers appealed to a student's sense of honor and decency. Unfortunately, these ideals were not deep in practice. The math teacher at the school would have students copy math problems from the chalkboard and then he would discipline any student who did not learn the lesson. In November, 1817, this math teacher beat a student so badly that the student died. After a protests from the students, the teacher was fired.
It was at this time that a new math teacher came to the school: Bernt Michael Homboe. He had a great love for mathematics that went beyond the standard mathematical education of the day in Norway. He assigned students independent projects. He was very impressed by the abilities of young Niels Abel and helped him to study advanced mathematical topics.
Niels began to greatly excel in mathematics. Despite this, many of the school's teachers were distressed by what was perceived as an overfocus on math. At this time, classical languages were considered the paradigm of a solid education. Abel would regularly visit the public library to study the works of Newton, Euler, Gauss, and Lagrange.
At the time, the University of Christiania offered major only in theology, medicine, or law. There was no track for mathematics. For this reason, Abel studied mathematics on his own. It was very clear that in order to continue his studies, he needed to go abroad. Owing mostly to a lack of funds, Abel stayed at the University of Christiania for 4 years.
In 1823, Abel moved to Denmark where he was able to meet the leading Nordic mathematician of the day, Ferdinand Degen. He lived with his aunt and her husband in Christianshavn. In his private studies, he made progress on number theory and elliptic equations. He also met his future fiancee Christine Kemp.
Abel's first proof was about the impossibility of finding a solution to the quintic equation which he published in 1824 when he was 22 years old. He translated the article to French and he condensed the proof to 6 pages. The resulting article was very difficult to read and was largely ignored by the leading mathematicians.
In 1824, he got a government grant to study mathematics abroad for 2 years. It was during these travels that he met an engineer named Leopold Crelle. In 1826, Crelle launched his soon-to-be-famous math magazine, Crelle's Journal. It was here that Abel would publish most of his work.
It was in Crelle's Journal, that Abel published an expanded version of his proof of the impossibility of solving the general quintic equation, a treatise on the binomial series, and six other submissions.
In July of 1826, Abel headed to Paris. His goal was to impress the very well established Paris Academy. He had worked hard to create a treatise on elliptic integrals. Known today as the Paris Treatise, this is a major mathematical breakthrough. In it, he establishes research areas that are still continuing today and he presented a generality of results that went beyond the mathematics of the day. He submitted this treatise to the Paris Academy in October of 1826. He never heard back. By the end of 1826, he headed back to Norway.
Once back, despite a lack of funds, Abel continued to make great advances in mathematics. He submitted a constant barrage of articles to Crelle's which was not able to publish them as quickly as Abel wrote them up.
In the Autumn of 1828, while Abel was making plans for his marriage, he became very sick and died. He was 26. A few weeks after he died, a letter from Crelle arrived offering Abel an important post in Berlin.
In 1841, the Paris Treatise was published in the Paris Academy's Journal.