Joseph Wedderburn
Encyclopedia
Joseph Henry Maclagan Wedderburn (2 February 1882, Forfar, Angus
, Scotland
– 9 October 1948, Princeton, New Jersey
) was a Scottish mathematician, who taught at Princeton University
for most of his career. A significant algebraist
, he proved that a finite division algebra
is a field
, and part of the Artin–Wedderburn theorem
on simple algebras. He also worked on group theory
and matrix algebra
.
. In 1903, he published his first three papers, worked as an assistant in the Physical Laboratory of the University, and obtained an M.A. degree with First Class Honours in mathematics.
He then studied briefly at the University of Leipzig
and the University of Berlin, where he met the algebraists Frobenius
and Schur
. A Carnegie Scholarship
allowed him to spend the 1904-1905 academic year at the University of Chicago
where he worked with Oswald Veblen
, E. H. Moore
, and most importantly, Leonard Dickson, who was to become the most important American algebraist of his day.
Returning to Scotland in 1905, Wedderburn worked for four years at the University of Edinburgh
as an assistant to George Chrystal
, who supervised his D.Sc, awarded in 1908 for a thesis titled On Hypercomplex Numbers. From 1906 to 1908, Wedderburn edited the Proceedings of the Edinburgh Mathematical Society. In 1909, he returned to the United States to become a Preceptor in Mathematics at Princeton University
; his colleagues included Luther P. Eisenhart
, Oswald Veblen
, Gilbert Ames Bliss
, and George Birkhoff.
Upon the outbreak of the First World War, Wedderburn enlisted in the British Army as a private. He was the first person at Princeton to volunteer for that war, and had the longest war service of anyone on the staff. While a Captain in the Fourth Field Survey Battalion in France, he devised sound-ranging equipment to locate enemy artillery.
He returned to Princeton after the war, becoming Associate Professor in 1921 and editing the Annals of Mathematics
until 1928. While at Princeton, he supervised only three Ph.Ds, one of them being Nathan Jacobson
. In his later years, Wedderburn became an increasingly solitary figure and may even have suffered from depression. His isolation after his 1945 early retirement was such that his death from a heart attack was not noticed for several days. His Nachlass
was destroyed, as per his instructions.
Wedderburn received the MacDougall-Brisbane Gold Medal and Prize from the Royal Society of Edinburgh in 1921, and was elected to the Royal Society of London in 1933.
As to why Wedderburn never married:
In 1905, Wedderburn published a paper that included three claimed proofs of a theorem stating that a noncommutative finite division ring
could not exist. The proofs all made clever use of the interplay between the additive group
of a finite division algebra
A, and the multiplicative group
A* = A-{0}. Parshall (1983) notes that the first of these three proofs had a gap not noticed at the time. Meanwhile, Wedderburn's Chicago colleague Dickson also found a proof of this result but, believing Wedderburn's first proof to be correct, Dickson acknowledged Wedderburn's priority. But Dickson also noted that Wedderburn constructed his second and third proofs only after having seen Dickson's proof. Parshall concludes that Dickson should be credited with the first correct proof.
A corollary to this theorem yields the complete structure of all finite projective geometry
. In their paper on "Non-Desarguesian and non-Pascalian geometries" in the 1907 Transactions of the American Mathematical Society
, Wedderburn and Veblen
showed that in these geometries, Pascal's theorem
is a consequence of Desargues' theorem
. They did so by constructing finite projective geometries which are neither "Desarguesian" nor "Pascalian" (the terminology is Hilbert
's).
Wedderburn's best-known paper was his sole-authored "On hypercomplex numbers," published in the 1907 Proceedings of the London Mathematical Society, and for which he was awarded the D.Sc. the following year. This paper gives a complete classification of simple and semisimple algebra
s. He then showed that every semisimple algebra
can be constructed as a direct sum of simple algebras and that every simple algebra is isomorphic to a matrix algebra
for some division ring
. The Artin–Wedderburn theorem
generalizes this result.
His best known book is his Lectures on Matrices (1934), which Jacobson praised as follows:
About Wedderburn's teaching:
Angus
Angus is one of the 32 local government council areas of Scotland, a registration county and a lieutenancy area. The council area borders Aberdeenshire, Perth and Kinross and Dundee City...
, Scotland
Scotland
Scotland is a country that is part of the United Kingdom. Occupying the northern third of the island of Great Britain, it shares a border with England to the south and is bounded by the North Sea to the east, the Atlantic Ocean to the north and west, and the North Channel and Irish Sea to the...
– 9 October 1948, Princeton, New Jersey
Princeton, New Jersey
Princeton is a community located in Mercer County, New Jersey, United States. It is best known as the location of Princeton University, which has been sited in the community since 1756...
) was a Scottish mathematician, who taught at Princeton University
Princeton University
Princeton University is a private research university located in Princeton, New Jersey, United States. The school is one of the eight universities of the Ivy League, and is one of the nine Colonial Colleges founded before the American Revolution....
for most of his career. A significant algebraist
Abstract algebra
Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras...
, he proved that a finite division algebra
Division algebra
In the field of mathematics called abstract algebra, a division algebra is, roughly speaking, an algebra over a field, in which division is possible.- Definitions :...
is a field
Field (mathematics)
In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
, and part of the Artin–Wedderburn theorem
Artin–Wedderburn theorem
In abstract algebra, the Artin–Wedderburn theorem is a classification theorem for semisimple rings. The theorem states that an Artinian semisimple ring R is isomorphic to a product of finitely many ni-by-ni matrix rings over division rings Di, for some integers ni, both of which are uniquely...
on simple algebras. He also worked on group theory
Group theory
In mathematics and abstract algebra, group theory studies the algebraic structures known as groups.The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces can all be seen as groups endowed with additional operations and...
and matrix algebra
Matrix algebra
Matrix algebra may refer to:*Matrix theory, is the branch of mathematics that studies matrices*Matrix ring, thought of as an algebra over a field or a commutative ring...
.
Life
Joseph Wedderburn was the tenth of fourteen children of Alexander Wedderburn, a physician, and Anne Ogilvie. In 1898, he entered the University of EdinburghUniversity of Edinburgh
The University of Edinburgh, founded in 1583, is a public research university located in Edinburgh, the capital of Scotland, and a UNESCO World Heritage Site. The university is deeply embedded in the fabric of the city, with many of the buildings in the historic Old Town belonging to the university...
. In 1903, he published his first three papers, worked as an assistant in the Physical Laboratory of the University, and obtained an M.A. degree with First Class Honours in mathematics.
He then studied briefly at the University of Leipzig
University of Leipzig
The University of Leipzig , located in Leipzig in the Free State of Saxony, Germany, is one of the oldest universities in the world and the second-oldest university in Germany...
and the University of Berlin, where he met the algebraists Frobenius
Ferdinand Georg Frobenius
Ferdinand Georg Frobenius was a German mathematician, best known for his contributions to the theory of differential equations and to group theory...
and Schur
Issai Schur
Issai Schur was a mathematician who worked in Germany for most of his life. He studied at Berlin...
. A Carnegie Scholarship
Carnegie Corporation of New York
Carnegie Corporation of New York, which was established by Andrew Carnegie in 1911 "to promote the advancement and diffusion of knowledge and understanding," is one of the oldest, largest and most influential of American foundations...
allowed him to spend the 1904-1905 academic year at the University of Chicago
University of Chicago
The University of Chicago is a private research university in Chicago, Illinois, USA. It was founded by the American Baptist Education Society with a donation from oil magnate and philanthropist John D. Rockefeller and incorporated in 1890...
where he worked with Oswald Veblen
Oswald Veblen
Oswald Veblen was an American mathematician, geometer and topologist, whose work found application in atomic physics and the theory of relativity. He proved the Jordan curve theorem in 1905.-Life:...
, E. H. Moore
E. H. Moore
Eliakim Hastings Moore was an American mathematician.-Life:Moore, the son of a Methodist minister and grandson of US Congressman Eliakim H. Moore, discovered mathematics through a summer job at the Cincinnati Observatory while in high school. He learned mathematics at Yale University, where he was...
, and most importantly, Leonard Dickson, who was to become the most important American algebraist of his day.
Returning to Scotland in 1905, Wedderburn worked for four years at the University of Edinburgh
University of Edinburgh
The University of Edinburgh, founded in 1583, is a public research university located in Edinburgh, the capital of Scotland, and a UNESCO World Heritage Site. The university is deeply embedded in the fabric of the city, with many of the buildings in the historic Old Town belonging to the university...
as an assistant to George Chrystal
George Chrystal
George Chrystal was a Scottish mathematician.He was born in Old Meldrum and educated at Aberdeen Grammar School and the University of Aberdeen, moving in 1872 to study under James Clerk Maxwell at Peterhouse, Cambridge, where he graduated Second wrangler in 1875, joint with William Burnside, and...
, who supervised his D.Sc, awarded in 1908 for a thesis titled On Hypercomplex Numbers. From 1906 to 1908, Wedderburn edited the Proceedings of the Edinburgh Mathematical Society. In 1909, he returned to the United States to become a Preceptor in Mathematics at Princeton University
Princeton University
Princeton University is a private research university located in Princeton, New Jersey, United States. The school is one of the eight universities of the Ivy League, and is one of the nine Colonial Colleges founded before the American Revolution....
; his colleagues included Luther P. Eisenhart
Luther P. Eisenhart
Luther Pfahler Eisenhart was an American mathematician, best known today for his contributions to semi-Riemannian geometry.-Life:...
, Oswald Veblen
Oswald Veblen
Oswald Veblen was an American mathematician, geometer and topologist, whose work found application in atomic physics and the theory of relativity. He proved the Jordan curve theorem in 1905.-Life:...
, Gilbert Ames Bliss
Gilbert Ames Bliss
Gilbert Ames Bliss, , was an American mathematician, known for his work on the calculus of variations.-Life:...
, and George Birkhoff.
Upon the outbreak of the First World War, Wedderburn enlisted in the British Army as a private. He was the first person at Princeton to volunteer for that war, and had the longest war service of anyone on the staff. While a Captain in the Fourth Field Survey Battalion in France, he devised sound-ranging equipment to locate enemy artillery.
He returned to Princeton after the war, becoming Associate Professor in 1921 and editing the Annals of Mathematics
Annals of Mathematics
The Annals of Mathematics is a bimonthly mathematical journal published by Princeton University and the Institute for Advanced Study. It ranks amongst the most prestigious mathematics journals in the world by criteria such as impact factor.-History:The journal began as The Analyst in 1874 and was...
until 1928. While at Princeton, he supervised only three Ph.Ds, one of them being Nathan Jacobson
Nathan Jacobson
Nathan Jacobson was an American mathematician....
. In his later years, Wedderburn became an increasingly solitary figure and may even have suffered from depression. His isolation after his 1945 early retirement was such that his death from a heart attack was not noticed for several days. His Nachlass
Nachlass
Nachlass is a German word, used in academia to describe the collection of manuscripts, notes, correspondence, and so on left behind when a scholar dies. The word is a compound in German: nach means 'after', and the verb lassen means 'leave'. The plural can be either Nachlasse or Nachlässe...
was destroyed, as per his instructions.
Wedderburn received the MacDougall-Brisbane Gold Medal and Prize from the Royal Society of Edinburgh in 1921, and was elected to the Royal Society of London in 1933.
As to why Wedderburn never married:
- "It seems that an old Scottish tradition required that a man, before marrying, accumulate savings equal to a certain percentage of his annual income. In Wedderburn's case his income had gone up so rapidly that he had never been able to accomplish this." (Hooke 1984)
Work
In all, Wedderburn published about 40 books and papers, making important advances in the theory of rings, algebras and matrix theory.In 1905, Wedderburn published a paper that included three claimed proofs of a theorem stating that a noncommutative finite division ring
Division ring
In abstract algebra, a division ring, also called a skew field, is a ring in which division is possible. Specifically, it is a non-trivial ring in which every non-zero element a has a multiplicative inverse, i.e., an element x with...
could not exist. The proofs all made clever use of the interplay between the additive group
Additive group
An additive group may refer to:*an abelian group, when it is written using the symbol + for its binary operation*a group scheme representing the underlying-additive-group functor...
of a finite division algebra
Division algebra
In the field of mathematics called abstract algebra, a division algebra is, roughly speaking, an algebra over a field, in which division is possible.- Definitions :...
A, and the multiplicative group
Multiplicative group
In mathematics and group theory the term multiplicative group refers to one of the following concepts, depending on the context*any group \scriptstyle\mathfrak \,\! whose binary operation is written in multiplicative notation ,*the underlying group under multiplication of the invertible elements of...
A* = A-{0}. Parshall (1983) notes that the first of these three proofs had a gap not noticed at the time. Meanwhile, Wedderburn's Chicago colleague Dickson also found a proof of this result but, believing Wedderburn's first proof to be correct, Dickson acknowledged Wedderburn's priority. But Dickson also noted that Wedderburn constructed his second and third proofs only after having seen Dickson's proof. Parshall concludes that Dickson should be credited with the first correct proof.
A corollary to this theorem yields the complete structure of all finite projective geometry
Projective geometry
In mathematics, projective geometry is the study of geometric properties that are invariant under projective transformations. This means that, compared to elementary geometry, projective geometry has a different setting, projective space, and a selective set of basic geometric concepts...
. In their paper on "Non-Desarguesian and non-Pascalian geometries" in the 1907 Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
Transactions of the American Mathematical Society is a monthly mathematics journal published by the American Mathematical Society. It started in 1900...
, Wedderburn and Veblen
Oswald Veblen
Oswald Veblen was an American mathematician, geometer and topologist, whose work found application in atomic physics and the theory of relativity. He proved the Jordan curve theorem in 1905.-Life:...
showed that in these geometries, Pascal's theorem
Pascal's theorem
In projective geometry, Pascal's theorem states that if an arbitrary hexagon is inscribed in any conic section, and pairs of opposite sides are extended until they meet, the three intersection points will lie on a straight line, the Pascal line of that configuration.- Related results :This theorem...
is a consequence of Desargues' theorem
Desargues' theorem
In projective geometry, Desargues' theorem, named in honor of Gérard Desargues, states:Denote the three vertices of one triangle by a, b, and c, and those of the other by A, B, and C...
. They did so by constructing finite projective geometries which are neither "Desarguesian" nor "Pascalian" (the terminology is Hilbert
David Hilbert
David Hilbert was a German mathematician. He is recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many areas, including invariant theory and the axiomatization of...
's).
Wedderburn's best-known paper was his sole-authored "On hypercomplex numbers," published in the 1907 Proceedings of the London Mathematical Society, and for which he was awarded the D.Sc. the following year. This paper gives a complete classification of simple and semisimple algebra
Semisimple algebra
In ring theory, a semisimple algebra is an associative algebra which has trivial Jacobson radical...
s. He then showed that every semisimple algebra
Semisimple algebra
In ring theory, a semisimple algebra is an associative algebra which has trivial Jacobson radical...
can be constructed as a direct sum of simple algebras and that every simple algebra is isomorphic to a matrix algebra
Matrix algebra
Matrix algebra may refer to:*Matrix theory, is the branch of mathematics that studies matrices*Matrix ring, thought of as an algebra over a field or a commutative ring...
for some division ring
Division ring
In abstract algebra, a division ring, also called a skew field, is a ring in which division is possible. Specifically, it is a non-trivial ring in which every non-zero element a has a multiplicative inverse, i.e., an element x with...
. The Artin–Wedderburn theorem
Artin–Wedderburn theorem
In abstract algebra, the Artin–Wedderburn theorem is a classification theorem for semisimple rings. The theorem states that an Artinian semisimple ring R is isomorphic to a product of finitely many ni-by-ni matrix rings over division rings Di, for some integers ni, both of which are uniquely...
generalizes this result.
His best known book is his Lectures on Matrices (1934), which Jacobson praised as follows:
- "That this was the result of a number of years of painstaking labour is evidenced by the bibliography of 661 items (in the revised printing) covering the period 1853 to 1936. The work is, however, not a compilation of the literature, but a synthesis that is Wedderburn's own. It contains a number of original contributions to the subject." (Nathan Jacobson, quoted in Taylor 1949)
About Wedderburn's teaching:
- "He was apparently a very shy man and much preferred looking at the blackboard to looking at the students. He had the galley proofGalley proofIn printing and publishing, proofs are the preliminary versions of publications meant for review by authors, editors, and proofreaders, often with extra wide margins. Galley proofs may be uncut and unbound, or in some cases electronic...
s from his book "Lectures on Matrices" pasted to cardboard for durability, and his "lecturing" consisted of reading this out loud while simultaneously copying it onto the blackboard." (Hooke 1984)
See also
- Wedderburn–Etherington number
- Wedderburn's little theoremWedderburn's little theoremIn mathematics, Wedderburn's little theorem states that every finite domain is a field. In other words, for finite rings, there is no distinction between domains, skew-fields and fields.The Artin–Zorn theorem generalizes the theorem to alternative rings....
- Wedderburn's theorem (division ring)Division ringIn abstract algebra, a division ring, also called a skew field, is a ring in which division is possible. Specifically, it is a non-trivial ring in which every non-zero element a has a multiplicative inverse, i.e., an element x with...
- Wedderburn's theorem (simple ring)Simple ringIn abstract algebra, a simple ring is a non-zero ring that has no ideal besides the zero ideal and itself. A simple ring can always be considered as a simple algebra. This notion must not be confused with the related one of a ring being simple as a left module over itself...