Nagell studied at the University of Oslo in Axel Thue. Then he visited several European universities (including Hamburg, Göttingen, Berlin, Strasbourg, Paris, Bologna, St. Petersburg). In 1931 he became a professor at Uppsala University, where he remained until his retirement in 1962. Even after that he lived in Uppsala.
During the Second World War, he supervised the Norwegian high school in Uppsala. In 1951 he was awarded the Norwegian St. Olaf's Medal ( Commendatore ), and in 1939 he became a Knight of the Swedish Order of the North Star. In 1947 he received the Cross of Liberty Haakon VII He was a member of the Norwegian Academy of Sciences, since 1952 the Kongelige Norske Videnskabers Selskab and since 1943 the Royal Swedish Academy of Sciences since 1925. Uppsala University awarded him in 1956 with an honorary doctorate.
Nagell was concerned with Diophantine equations (solutions of polynomials in whole numbers ). Specifically, he focused on elliptic curves over the rational numbers (curves of genus 1 ). He is best known for the set of Lutz and Nagell, which allows a calculation of the Torsionspunkte elliptic curves over the rational numbers. He also examined the possible Torsionsgruppen. Here he was, however, overtaken by the results of Barry Mazur 1977.
He was married twice and had two children from his first marriage.
- Introduction to Number Theory, Wiley 1951, 2nd edition Chelsea 1981.
- L'analyze de indéterminée degré supérieur, Gauthier -Villars, Paris 1929
- Collected Papers. , Queen 's University, Queen's papers in pure and applied mathematics, 4 volumes, 2002 ( editor Paulo Ribenboim ).