KUMMER, ERNST EDUARD

born Jan. 29, 1810, Sorau, Brandenburg, Prussia died May 14, 1893, Berlin, Ger. German mathematician whose introduction of ideal numbers, which are defined as a special subgroup of a ring, extended the fundamental theorem of arithmetic to complex number fields. After teaching in gymnasiums one year at Sorau and 10 years at Liegnitz (now Legnica, Pol.), Kummer became professor of mathematics at the university at Breslau (now Wroclaw, Pol.) in 1842. In 1855 he succeeded Peter Gustav Lejeune Dirichlet as professor of mathematics at the University of Berlin, at the same time also becoming professor at the Berlin War College. In 1843 Kummer showed to Dirichlet an attempted proof of Fermat's last theorem (xn + yn = zn, where n is an integer greater than 2, has no solution for positive integral values of x, y, and z); Dirichlet found an error. Kummer continued his search and developed the concept of ideal numbers. Using this concept he proved the insolubility of the Fermat relation for all but a small group of primes, and he thus laid the foundation for an eventual complete proof of Fermat's last theorem. For his great advance the Paris Academy of Sciences awarded him the Grand Prize in 1857. The ideal numbers have made possible new developments in the arithmetic of algebraic numbers. Inspired by the work of William Hamilton on systems of optical rays, Kummer developed the surface named in his honour, based on the quartic (fourth-power) equation that is the singular surface of the quadratic line complex. This surface is the wave surface in space of four dimensions. Kummer also extended the work of Carl Gauss on the hypergeometric series, adding developments that are useful in the theory of differential equations.