Niels Abel proved the impossibility of solving algebraically the general equation of the fifth degree.

Abel's life was dominated by poverty. After the death of his father, who was a Protestant minister, in 1820 Abel had the responsibility of supporting his mother and family.

Abel's teacher Holmboe, recognising his talent for mathematics, raised money from his colleagues to enable Abel to attend Christiania University. He entered the university in 1821, 10 years after the university was founded, and graduated in 1822.

Abel published in 1823 papers on functional equations and integrals. In it Abel gives the first solution of an integral equation. In 1824 he proved the impossibility of solving algebraically the general equation of the fifth degree and published it at his own expense hoping to obtain recognition for his work.

He eventually won a scholarship from the Norwegian government to travel abroad, visiting Germany and France.

Abel was instrumental in establishing mathematical analysis on a rigorous basis. His major work Recherches sur les fonctions elliptiques was published in 1827 in the first volume of Crelle's Journal, the first journal devoted entirely to mathematics. Abel met Crelle on his visit to Germany.

After visiting Paris he returned to Norway heavily in debt. While in Paris he visited a doctor who informed him he had tuberculosis. Despite his bad health and poverty he continued writing papers on equation theory and elliptic functions of major importance in the development of the whole theory. Abel revolutionised the understanding of elliptic functions by studying the inverse of these functions.

Abel travelled by sled to visit his fianceé for Christmas 1828 in Froland. He became seriously ill on the sled journey and died a couple of months later.