Øystein Ore

Ore is known for his work in ring theory, Galois connections, and most of all, graph theory.

His early work was on algebraic number fields, how to decompose the ideal generated by a prime number into prime ideals. He then worked on noncommutative rings, proving his celebrated theorem on embedding a domain into a division ring. He then examined polynomial rings over skew fields, and attempted to extend his work on factorisation to non-commutative rings. The Ore condition, which (if true) allows a ring of fractions to be defined, and the Ore extension, a non-commutative analogue of rings of polynomials, are part of this work. In more elementary number theory, Ore's harmonic numbers are the numbers whose divisors have an integer harmonic mean.

As a teacher, Ore is notable for supervising two doctoral students who would make contributions to science and mathematics: Grace Hopper, who eventually became a United States rear admiral and computer scientist and who was a pioneer in developing the first computers, and Marshall Hall, Jr., an American mathematician who did important research in group theory and combinatorics.

In 1930, the Collected Works of Richard Dedekind were published in three volumes, jointly edited by Ore and Emmy Noether. He then turned his attention to lattice theory becoming, together with Garrett Birkhoff, one of the two founders of American expertise in the subject. Ore's early work on lattice theory led him to the study of equivalence and closure relations, Galois connections, and finally to graph theory, which occupied him to the end of his life. He wrote two books on the subject, one on the theory of graphs and another on their applications. Within graph theory, Ore's theorem is one of several results proving that sufficiently dense graphs contain Hamiltonian cycles.

Ore had a lively interest in the history of mathematics, and was an unusually able author of books for laypeople, such as his biographies of Cardano and Niels Henrik Abel.