Wedderburn, Joseph Henry Maclagan (1882–1948)

Scottish mathematician who opened new lines of thought in the subject of mathematical fields and who had a deep influence on the development of modern algebra.

Life Wedderburn was born in Forfar and studied at Edinburgh and Chicago, USA. He taught in the USA at Princeton 1909–45, though during World War I he saw active duty in France as a soldier in the British army.

Hyper-complex numbers The first paper that Wedderburn published, ‘Theorem on Finite Algebra’ (1905), was a milestone in algebraic history. By introducing new methods, he showed that it was possible to arrive at total understanding of the structure of semi-simple algebras using hyper-complex numbers as well as real or complex numbers.

Wedderburn went on to derive the two theorems to which his name has become attached. The first was contained in his paper ‘On Hyper-Complex Numbers’ (1907), in which he demonstrated that a simple algebra consists of matrices of a given degree with elements taken from a division of algebra.

The first Wedderburn theorem states that ‘if the algebra is a finite division algebra (that is, that it has only a finite number of elements and always permits division by a non-zero element), then the multiplication law must be commutative, so that the algebra is actually a finite field’.

Fields with a finite number of elements Wedderburn's second theorem states that a central-simple algebra is isomorphic to the algebra of all n × n algebras. He arrived at it by an investigation of skew fields with a finite number of elements.

His discovery that every field with a finite number of elements is commutative under multiplication led to a complete classification of all semi-simple algebras with a finite number of elements.