Algebras

At the end of the nineteenth century some major review works appeared. The German David Hilbert (1862–1943) published in 1897 a long report on algebraic number theory. The next year the Englishman Alfred North Whitehead (1861–1947) put out a detailed summary of several of them in his large book A Treatise on Universal Algebra, inspired by Grassmann but covering also Boole's logic, aspects of geometries, linear algebra, vectors, and parts of applied mathematics; an abandoned sequel was to have included quaternions. His title, taken from Sylvester, was not happy: no algebra is universal in the sense of embracing all others, and Whitehead did not offer one.

Elsewhere, group theory rose further in status, to be joined by other abstract algebras, such as rings, fields (already recognized by Abel and Galois in their studies of polynomial equations), ideals, integral domains, and lattices, each inspired by applications. German-speaking mathematicians were especially prominent, as was the rising new mathematical nationality, the Americans. Building upon the teaching of Emmy Noether (1882–1935) and Emil Artin (1898–1962), B. L. van der Waerden's (1903–1996) book Modern Algebra became a standard text for abstract algebras and several applications, from the first (1930–1931) of its many editions.

This abstract approach solved the mystery of the need for complex numbers when finding real roots of real polynomial equations. The key notion is closure: an algebra A is closed relative to an operation O on its objects a, or to a means of combining a and b, if Oa and a·b always belong to A. Now finding roots involved the operations of taking square, cube, … roots and complex but not real numbers are closed relative to them.

One of the most striking features of mathematics in the twentieth century was the massive development of topology. Algebraic topology and topological groups are two of its parts, and algebras of various kinds have informed several others. Both (abstract) algebras and topology featured strongly in the formalization of pure mathematics expounded mainly after World War II by a team of French mathematicians writing under the collective name "Bourbaki."