Øystein Ore
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Øystein Ore (7 October 1899 – 13 August 1968) was a Norwegian
mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, mathematical structure, structure, space, Mathematica ...
known for his work in ring theory,
Galois connection In mathematics, especially in order theory, a Galois connection is a particular correspondence (typically) between two partially ordered sets (posets). Galois connections find applications in various mathematical theories. They generalize the fun ...
s,
graph theory In mathematics and computer science, graph theory is the study of ''graph (discrete mathematics), graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of ''Vertex (graph ...
, and the
history of mathematics The history of mathematics deals with the origin of discoveries in mathematics and the History of mathematical notation, mathematical methods and notation of the past. Before the modern age and the worldwide spread of knowledge, written examples ...
.


Life

Ore graduated from the
University of Oslo The University of Oslo (; ) is a public university, public research university located in Oslo, Norway. It is the List of oldest universities in continuous operation#Europe, oldest university in Norway. Originally named the Royal Frederick Univ ...
in 1922, with a Cand.Real.degree in mathematics. In 1924, the University of Oslo awarded him the Ph.D. for a thesis titled ''Zur Theorie der algebraischen Körper'', supervised by
Thoralf Skolem Thoralf Albert Skolem (; 23 May 1887 – 23 March 1963) was a Norwegian mathematician who worked in mathematical logic and set theory. Life Although Skolem's father was a primary school teacher, most of his extended family were farmers. Skole ...
. Ore also studied at
Göttingen University Göttingen (, ; ; ) is a college town, university city in Lower Saxony, central Germany, the Capital (political), capital of Göttingen (district), the eponymous district. The River Leine runs through it. According to the 2022 German census, t ...
, where he learned
Emmy Noether Amalie Emmy Noether (23 March 1882 – 14 April 1935) was a German mathematician who made many important contributions to abstract algebra. She also proved Noether's theorem, Noether's first and Noether's second theorem, second theorems, which ...
's new approach to
abstract algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures, which are set (mathematics), sets with specific operation (mathematics), operations acting on their elements. Algebraic structur ...
. He was also a fellow at the Mittag-Leffler Institute in
Sweden Sweden, formally the Kingdom of Sweden, is a Nordic countries, Nordic country located on the Scandinavian Peninsula in Northern Europe. It borders Norway to the west and north, and Finland to the east. At , Sweden is the largest Nordic count ...
, and spent some time at the
University of Paris The University of Paris (), known Metonymy, metonymically as the Sorbonne (), was the leading university in Paris, France, from 1150 to 1970, except for 1793–1806 during the French Revolution. Emerging around 1150 as a corporation associated wit ...
. In 1925, he was appointed research assistant at the University of Oslo.
Yale University Yale University is a Private university, private Ivy League research university in New Haven, Connecticut, United States. Founded in 1701, Yale is the List of Colonial Colleges, third-oldest institution of higher education in the United Stat ...
’s James Pierpont went to Europe in 1926 to recruit research mathematicians. In 1927, Yale hired Ore as an assistant professor of mathematics, promoted him to associate professor in 1928, then to full professor in 1929. In 1931, he became a
Sterling Professor Sterling Professor, the highest academic rank at Yale University, is awarded to a Academic tenure in North America, tenured faculty member considered the best in their field. It is akin to the rank of distinguished professor at other universities. ...
(Yale's highest academic rank), a position he held until he retired in 1968. Ore gave an
American Mathematical Society The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, ...
Colloquium lecture in 1941 and was a plenary speaker at the International Congress of Mathematicians in 1936 in Oslo. He was also elected to the
American Academy of Arts and Sciences The American Academy of Arts and Sciences (The Academy) is one of the oldest learned societies in the United States. It was founded in 1780 during the American Revolution by John Adams, John Hancock, James Bowdoin, Andrew Oliver, and other ...
and the Oslo Academy of Science. He was a founder of the
Econometric Society The Econometric Society is an international society of academic economists interested in applying statistical tools in the practice of econometrics. It is an independent organization with no connections to societies of professional mathematicians o ...
. Ore visited Norway nearly every summer. During
World War II World War II or the Second World War (1 September 1939 – 2 September 1945) was a World war, global conflict between two coalitions: the Allies of World War II, Allies and the Axis powers. World War II by country, Nearly all of the wo ...
, he was active in the "American Relief for Norway" and "Free Norway" movements. In gratitude for the services rendered to his native country during the war, he was decorated in 1947 with the
Order of St. Olav The Royal Norwegian Order of Saint Olav (; or ''Sanct Olafs Orden'', the old Norwegian name) is a Norwegian order of chivalry instituted by King Oscar I on 21 August 1847. It is named after King Olav II, known to posterity as St. Olav. Just be ...
. In 1930, Ore married Gudrun Lundevall. They had two children. Ore had a passion for painting and sculpture, collected ancient maps, and spoke several languages.


Work

Ore is known for his work in ring theory, Galois connections, and most of all,
graph theory In mathematics and computer science, graph theory is the study of ''graph (discrete mathematics), graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of ''Vertex (graph ...
. His early work was on
algebraic number field In mathematics, an algebraic number field (or simply number field) is an extension field K of the field of rational numbers such that the field extension K / \mathbb has finite degree (and hence is an algebraic field extension). Thus K is a ...
s, how to decompose the ideal generated by a
prime number A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...
into
prime ideal In algebra, a prime ideal is a subset of a ring (mathematics), ring that shares many important properties of a prime number in the ring of Integer#Algebraic properties, integers. The prime ideals for the integers are the sets that contain all th ...
s. He then worked on
noncommutative ring In mathematics, a noncommutative ring is a ring whose multiplication is not commutative; that is, there exist ''a'' and ''b'' in the ring such that ''ab'' and ''ba'' are different. Equivalently, a ''noncommutative ring'' is a ring that is not ...
s, proving his celebrated theorem on embedding a domain into a
division ring In algebra, a division ring, also called a skew field (or, occasionally, a sfield), is a nontrivial ring in which division by nonzero elements is defined. Specifically, it is a nontrivial ring in which every nonzero element has a multiplicativ ...
. He then examined
polynomial ring In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring formed from the set of polynomials in one or more indeterminates (traditionally also called variables) with coefficients in another ring, ...
s over
skew field In algebra, a division ring, also called a skew field (or, occasionally, a sfield), is a nontrivial ring in which division by nonzero elements is defined. Specifically, it is a nontrivial ring in which every nonzero element has a multiplicative ...
s, 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 Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example ...
, Ore's harmonic numbers are the numbers whose
divisor In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a '' multiple'' of m. An integer n is divisible or evenly divisibl ...
s have an integer
harmonic mean In mathematics, the harmonic mean is a kind of average, one of the Pythagorean means. It is the most appropriate average for ratios and rate (mathematics), rates such as speeds, and is normally only used for positive arguments. The harmonic mean ...
. As a teacher, Ore is notable for supervising two doctoral students who would make contributions to science and mathematics:
Grace Hopper Grace Brewster Hopper (; December 9, 1906 – January 1, 1992) was an American computer scientist, mathematician, and United States Navy rear admiral. She was a pioneer of computer programming. Hopper was the first to devise the theory of mach ...
, who eventually became a
United States The United States of America (USA), also known as the United States (U.S.) or America, is a country primarily located in North America. It is a federal republic of 50 U.S. state, states and a federal capital district, Washington, D.C. The 48 ...
rear admiral Rear admiral is a flag officer rank used by English-speaking navies. In most European navies, the equivalent rank is called counter admiral. Rear admiral is usually immediately senior to commodore and immediately below vice admiral. It is ...
and
computer scientist A computer scientist is a scientist who specializes in the academic study of computer science. Computer scientists typically work on the theoretical side of computation. Although computer scientists can also focus their work and research on ...
and who was a pioneer in developing the first
computer A computer is a machine that can be Computer programming, programmed to automatically Execution (computing), carry out sequences of arithmetic or logical operations (''computation''). Modern digital electronic computers can perform generic set ...
s, and Marshall Hall, Jr., an American mathematician who did important research in
group theory In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ( ...
and
combinatorics Combinatorics is an area of mathematics primarily concerned with counting, both as a means and as an end to obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many ...
. In 1930, the ''Collected Works of
Richard Dedekind Julius Wilhelm Richard Dedekind (; ; 6 October 1831 – 12 February 1916) was a German mathematician who made important contributions to number theory, abstract algebra (particularly ring theory), and the axiomatic foundations of arithmetic. H ...
'' were published in three volumes, jointly edited by Ore and
Emmy Noether Amalie Emmy Noether (23 March 1882 – 14 April 1935) was a German mathematician who made many important contributions to abstract algebra. She also proved Noether's theorem, Noether's first and Noether's second theorem, second theorems, which ...
. He then turned his attention to
lattice theory A lattice is an abstract structure studied in the mathematical subdisciplines of order theory and abstract algebra. It consists of a partially ordered set in which every pair of elements has a unique supremum (also called a least upper bou ...
becoming, together with
Garrett Birkhoff Garrett Birkhoff (January 19, 1911 – November 22, 1996) was an American mathematician. He is best known for his work in lattice theory. The mathematician George Birkhoff (1884–1944) was his father. Life The son of the mathematician Ge ...
, 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 connection In mathematics, especially in order theory, a Galois connection is a particular correspondence (typically) between two partially ordered sets (posets). Galois connections find applications in various mathematical theories. They generalize the fun ...
s, and finally to
graph theory In mathematics and computer science, graph theory is the study of ''graph (discrete mathematics), graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of ''Vertex (graph ...
, 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 cycle In the mathematics, mathematical field of graph theory, a Hamiltonian path (or traceable path) is a path (graph theory), path in an undirected or directed graph that visits each vertex (graph theory), vertex exactly once. A Hamiltonian cycle (or ...
s. Ore had a lively interest in the
history of mathematics The history of mathematics deals with the origin of discoveries in mathematics and the History of mathematical notation, mathematical methods and notation of the past. Before the modern age and the worldwide spread of knowledge, written examples ...
, and was an unusually able author of books for laypeople, such as his biographies of Cardano and
Niels Henrik Abel Niels Henrik Abel ( , ; 5 August 1802 – 6 April 1829) was a Norwegian mathematician who made pioneering contributions in a variety of fields. His most famous single result is the first complete proof demonstrating the impossibility of solvin ...
.


Books by Ore

* ''Les Corps Algébriques et la Théorie des Idéaux'' (1934) * ''L'Algèbre Abstraite'' (1936) *
Number Theory and its History
' (1948) * ''Cardano, the Gambling Scholar'' (Princeton University Press, 1953) * ''Niels Henrik Abel, Mathematician Extraordinary'' (U. of Minnesota Press, 1957) * ''Theory of Graphs'' (1962) * ''Graphs and Their Uses'' (1963) * ''The Four-Color Problem'' (1967) * ''Invitation to Number Theory'' (1969)


Articles by Ore

* * * * * * * * *


See also

* Deficiency (graph theory) * Geodetic graph *
Magma (algebra) In abstract algebra, a magma, binar, or, rarely, groupoid is a basic kind of algebraic structure. Specifically, a magma consists of a Set (mathematics), set equipped with a single binary operation that must be closure (binary operation), closed b ...
*
Ore algebra In computer algebra, an Ore algebra is a special kind of iterated Ore extension that can be used to represent linear functional operators, including linear differential and/or recurrence operators. The concept is named after Øystein Ore. Def ...
* Ore condition * Ore's conjecture * Ore extension * Ore number * Ore's theorem * Schwartz–Zippel lemma *
Universal algebra Universal algebra (sometimes called general algebra) is the field of mathematics that studies algebraic structures in general, not specific types of algebraic structures. For instance, rather than considering groups or rings as the object of stud ...


References


External links

* . The source for much of this entry. {{DEFAULTSORT:Ore, Oeystein 20th-century Norwegian mathematicians Combinatorialists Lattice theorists Yale University faculty Historians of mathematics 1899 births 1968 deaths Yale Sterling Professors