BAKER, ALAN

born Aug. 19, 1939, London, Eng. British mathematician who was awarded the Fields Medal in 1970 for his work in number theory. Baker attended University College, London (B.S., 1961), and Trinity College, Cambridge (M.A. and Ph.D., 1964). He held an appointment at University College (196465) and then joined the faculty of Trinity College in 1966. Baker received the Fields Medal at the International Congress of Mathematicians in Nice, France, in 1970. His work with diophantine equations provided an advance over previous work in an area that a few years earlier had been shown to hold limited possibilities for success. Following work by Axel Thue, Carl Ludwig Siegel, and Klaus Friedrich Roth, Baker showed that for f(x, y) = m, f(x, y), an irreducible binary form of degree n 3 with integer coefficients, m being a positive integer, there is an effective bound B depending only on n and on the coefficients of f, so that max(| x0 |, | y0 |) B, for any solution (x0, y0). Thus, at least in theory, it is possible to determine all the solutions explicitly for a large class of equations. This work was related to Baker's considerable generalization of the Gelfond-Schneider theorem (Hilbert's seventh problem) that states that, if a and b are algebraic, a 0, 1, and b is irrational, then ab is transcendental. Baker's generalization states that, if a1, , ak ( 0, 1) are algebraic and if 1, b1, , bk are linearly independent over the rationals and if all the bi are irrational algebraic numbers, then a1b1akbk is transcendental. Paul Turn remarked in his description of Baker's work in the proceedings of the Nice Congress that his achievement was made all the more impressive by Hilbert's having predicted that the problem posed by the Riemann hypothesis, which remains unproved, would be settled long before the proof of the transcendence of ab. Baker's publications include Transcendental Number Theory (1975).