Math Labs

Alan Baker

1939–2018, London, England, Cambridge, England

Arithmetic and number theoryFields Medal 1970

British mathematician whose effective estimates for linear forms in logarithms revolutionized transcendence theory and made Diophantine equations effectively solvable; Fields Medal 1970.

Alan Baker was born in London and studied at University College London and Trinity College, Cambridge, where he took his doctorate under Harold Davenport. In 1966–1967 he proved his celebrated theorem on linear forms in logarithms: if α1,…,αn\alpha_1,\dots,\alpha_n are algebraic numbers whose logarithms are linearly independent over the rationals, then any nonvanishing form β0+β1log⁡α1+⋯+βnlog⁡αn\beta_0 + \beta_1\log\alpha_1 + \cdots + \beta_n\log\alpha_n with algebraic coefficients satisfies an explicit lower bound depending only on the heights and degrees of the numbers involved.

The consequences were immediate and far-reaching. Baker obtained effective bounds for the solutions of the Catalan conjecture, of unit equations and of Thue equations, gave effective versions of class-number results for imaginary quadratic fields, and helped settle Gauss's class-number-one problem. His book "Transcendental Number Theory" (1975) became the standard reference for the field he had reshaped.

Baker spent nearly his whole career at Cambridge, becoming a Fellow of Trinity College and professor of pure mathematics. He received the Fields Medal in 1970. He died in Cambridge in 2018.

Workplaces: University of Cambridge

United Kingdom

Award-winning work

Fields Medal 1970Awarded for his work on Diophantine analysis, especially his theorem giving effective lower bounds for nonvanishing linear forms in logarithms of algebraic numbers. The result transformed transcendence theory: it gave an effective solution to the question underlying Hilbert's seventh problem and yielded effective bounds for the solutions of many classical Diophantine equations, including the Catalantype equation and unit equations. Building on and greatly extending Gelfond's and Schneider's methods, Baker's estimates made large parts of number theory effective for the first time and remain an indispensable tool in Diophantine approximation.

Contributions, linked to the library

Great problemHilbert's seventh problemHis linear-forms-in-logarithms bounds gave effective versions of the transcendence results answering Hilbert's seventh problem.Great problemCatalan's conjectureBaker's bounds gave the first effective finiteness results for the equation xp−yq=1x^p - y^q = 1, later solved outright by Mihăilescu.