Joseph Henry Maclagan Wedderburn, M.A., D.Sc., F.R.S.

by H W Turnbull

Joseph Henry Maclagan Wedderburn, Mathematician, Professor Emeritus of Princeton University, U.S.A., died at his home in Princeton on October 9, 1948, at the age of sixty-six.

Professor Wedderburn, whose brother is our Treasurer, Sir Ernest Wedderburn, was born at Forfar on February 26, 1882, and educated in Edinburgh, where he graduated in 1903.

While still an undergraduate, he was elected in 1902 to membership of the Edinburgh Mathematical Society, and soon afterwards in 1903 to the fellowship of the Royal Society of Edinburgh. In 1921 he was awarded the Makdougall-Brisbane Prize for his researches into Hypercomplex Numbers. From 1904 to 1909 he assisted Professor George Chrystal at Edinburgh, where he received the degree of D.Sc. in 1908, and is still remembered for introducing the course on general analysis; but the natural bent of his mind was toward abstract algebra. This interest had been fostered by post-graduate residence in Leipzig, Berlin and Chicago, especially the last, where the stimulating teaching of E H Moore and L E Dickson brought him permanent inspiration.

In 1909 he left Edinburgh and joined the Faculty of Mathematics at Princeton University. During the First World War, which interrupted his career for five years, he served with the Seaforth Highlanders, volunteering with characteristic modesty as a private and rising to the rank of captain at the end of the war: he became 0. C. of an Observation Group in the Royal Engineers, and was mentioned in dispatches.

At Princeton, Wedderburn took part in all the activities of the Department of Mathematics. That he was an exceptionally good teacher has been the testimony of generations of undergraduates. The present eminence of American mathematicians in algebra owes not a little to his influence in the classroom and to his editorship-in-chief, for twenty years, of the Annals of Mathematics, dating from the time in 1911 when the Department of Mathematics took charge of this periodical, which steadily rose to its present distinguished position among the mathematical journals of the world. The volume of the Annals published in 1946, which bears an excellent portrait of Wedderburn, was in effect a Festschrift dedicated to him. This may be taken as an indication of the respect in which he was held, not only in America, but by mathematicians throughout the world. In 1933 he was elected Fellow of the Royal Society.

His colleague, Professor Alan W C Menzies, writes that Wedderbum had been in poor health for a number of years. "He had an excellent tenor voice and would, if in the mood, sing a dozen or more songs from Songs of the North, or fewer from Mrs Kennedy Fraser's collection of Hebridean songs. ... As a hobby he at one time wove on a handloom in his attic." He never married, and was accustomed to spend his vacations in a remote farm in the Adirondacks.

While Wedderbum had a broad knowledge of pure and applied mathematics, his research work was exclusively done in algebra. Indeed algebra was his all-embracing passion, and to algebra he made first-rate contributions, distinguished no less by their originality than by their depth. He was not a prolific writer, but everything that he wrote was significant and marked with the seal of unmistakable originality. This gives an impression of reserve and distinction to his work.

The two earliest of his publications occur in the Transactions of the Royal Society of Edinburgh (XL, 1903) and the Proceedings of the Edinburgh Mathematical Society of the next year. But the most important of all his works, to judge in the light of its great influence upon abstract algebra, is his memoir On Hypercomplex Numbers that appeared in the Proceedings of the London Mathematical Society (2), VI (1906), 77-119. Thereafter he published his work in America. Hypercomplex numbers start with the discoveries of Hamilton and Grassmann a century ago: they are developments of the complex number x + iy, which has two components x, iy, to higher categories of number which have a plurality of components: further, these components and these resultant numbers need no longer commute, namely, the property ab=ba is not assumed always to hold. Cayley, Sylvester, Pierce, Molien, Cartan, Dickson and Frobenius had developed the theory to high perfection during the sixty years prior to Wedderbum's memoir: yet there was something essential still to be done. The products ab and ba may or may not be equal: what then are the conditions, for a given a, that they are equal? This question could be answered, but the answer involved solving an equation which might be insoluble in terms of the number field under discussion. For example, if real quaternions are studied it is a blemish to solve a problem by having to use not real but complex quaternions. Wedderburn removed this blemish and revolutionised the theory by making it fit the appropriate field. His technique, of using complexes (analogous to whole lines, whole planes and so on, and not merely points, in geometry), advanced and enriched this algebra and made the deeper treatment possible. He did for linear associative algebra both what Kronecker had done for matrices by developing the rational treatment, and also what Frobenius had done for finite groups by using complexes.

Most characteristic is his theorem (1906) that any simple algebra can be expressed as the direct product of a primitive (i.e. a division) algebra and a simple matrix algebra. As he tells us, this result was implicit in a classical work by Cartan published nine years earlier but in an obscure form, and then only for the case of the complex field. To have seen the true significance of Cartan's result and to have put it into a fruitful context, alone would have been a fine achievement: but to generalise it so as to hold for any field was a triumph. Wedderburn's theorem has become a classic. Furthermore, he carried the study of algebraic structure to new heights by separating out the singular part of a hypercomplex number system from the ordinary part; that is, by proving that such a system could always be partitioned into a radical together with an aggregate of simple algebras, the radical consisting entirely of properly nilpotent elements. This step may be compared with that of Kronecker when he accounted for singular pencils in the theory of matrices. As a purely mathematical investigation his whole theory is wonderful: it has taken on new and rich significance with the rise of quantum mechanics.

Wedderbum's book, Lectures on Matrices (1934), which deals with all these matters, is that of a master. It is not easy reading, but clearly it exhibits the contribution of Edinburgh through Tait and Knott, based upon the quaternions of Hamilton, as a distinctive pattern in a many-sided theory that has been enriched by the Continent and America.

The handloom in his attic and the tartan of the clan are the true symbols of the precise and coloured design that Wedderbum discovered in abstract algebra.

Joseph Henry Maclagan Wedderburn's RSE obituary by H W Turnbull appeared in Royal Society of Edinburgh Year Book 1948 and 1949, 51-52.

See also Obituary Notices of Fellows of the Royal Society, vi, 1948-49, pp. 619-626.