**Joseph Henry Maclagan Wedderburn** FRSE FRS (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.^{[2]}^{[3]}

His younger brother was the lawyer Ernest Wedderburn.

## Contents

## Life

Joseph Wedderburn was the tenth of fourteen children of Alexander Wedderburn of Pearsie, a physician, and Anne Ogilvie. He was educated at Forfar Academy then in 1895 his parents sent Joseph and his younger brother Ernest to live in Edinburgh with their paternal uncle, J R Maclagan Wedderburn, allowing them to attend George Watson's College. This house was at 3 Glencairn Crescent in the West End of the city.^{[4]}

In 1898 Joseph entered the University of Edinburgh. In 1903, he published his first three papers, worked as an assistant in the Physical Laboratory of the University, obtained an MA degree with First Class Honours in mathematics, and was elected a Fellow of the Royal Society of Edinburgh, upon the proposal of George Chrystal, James Gordon MacGregor, Cargill Gilston Knott and William Peddie. Aged only 21 he remains one of the youngest Fellows ever.^{[5]}

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*. He gained a PhD in algebra from the University of Edinburgh in 1908.^{[6]} 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. He served with the Seaforth Highlanders in France, as Lieutenant (1914), then as Captain of the 10th Battalion (1915–18). While a Captain in the Fourth Field Survey Battalion of the Royal Engineers 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 PhDs, 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.^{[1]}

## 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 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 also constructed 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 algebras. He then showed that every semisimple algebra finite-dimensional 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 generalises this result, with the ascending chain condition.

His best known book is his *Lectures on Matrices* (1934),^{[7]} 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 proofs 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

- Hypercomplex numbers
- Wedderburn–Etherington number
- Wedderburn's little theorem
- Wedderburn's theorem (division ring)
- Wedderburn's theorem (simple ring)

## References

- ↑
^{1.0}^{1.1}Taylor, H. S. (1949). "Joseph Henry Maclagen Wedderburn. 1882-1948". pp. 618–626. Digital object identifier:10.1098/rsbm.1949.0016. JSTOR 768943. - ↑ O'Connor, John J.; Robertson, Edmund F.. "MacTutor History of Mathematics archive". University of St Andrews..
- ↑ J at the Mathematics Genealogy Project
- ↑ Edinburgh Post Office Directory 1895
- ↑
*Biographical Index of Former Fellows of the Royal Society of Edinburgh 1783–2002*. The Royal Society of Edinburgh. July 2006. ISBN 978-0-902198-84-5. https://www.royalsoced.org.uk/cms/files/fellows/biographical_index/fells_indexp2.pdf. - ↑ J.H., Maclagan-Wedderburn (1908). "Theory of linear associative algebras" (in en).
- ↑ MacDuffee, C. C. (1935). "Wedderburn on Matrices". pp. 471–472. Digital object identifier:10.1090/s0002-9904-1935-06117-8. http://projecteuclid.org/euclid.bams/1183498283.

## Further reading

- Artin, Emil (1950). "The influence of J. H. M. Wedderburn on the development of modern algebra". pp. 65–72. Digital object identifier:10.1090/s0002-9904-1950-09346-x. http://projecteuclid.org/euclid.bams/1183514454.
- Robert Hooke (1984) Recollections of Princeton, 1939–1941
- Karen Parshall (1983) "In pursuit of the finite division algebra theorem and beyond: Joseph H M Wedderburn, Leonard Dickson, and Oswald Veblen,"
*Archives of International History of Science 33*: 274–99. - Karen Parshall (1985) "Joseph H. M. Wedderburn and the structure theory of algebras,"
*Archive for History of Exact Sciences 32*: 223–349. - Karen Parshall (1992) "New Light on the Life and Work of Joseph Henry Maclagan Wedderburn (1882–1948)," in Menso Folkerts
*et al.*(eds.):*Amphora: Festschrift für Hans Wußing zu seinem 65. Geburtstag*, Birkhäuser Verlag, 523–537.

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