Alan Baker

Alan Baker
Born	8/19/1939
Birthplace	London, England
Died	2/4/2018 Cambridge, England
Nationality	British
Occupation	Mathematician
Known for	Work on transcendental number theory, Baker's theorem
Education	PhD, University of Cambridge
Awards	Fields Medal (1970), Adams Prize, Fellow of the Royal Society

Alan Baker (19 August 1939 – 4 February 2018) was a British mathematician whose contributions to number theory earned him the Fields Medal in 1970, one of the highest honours in mathematics. Born in London, Baker spent much of his academic career at the University of Cambridge, where he produced foundational work on transcendental numbers and Diophantine equations. His results on linear forms in logarithms of algebraic numbers, now commonly known as Baker's theorem, represented a major advance in the understanding of transcendental number theory and provided effective methods for solving a wide class of number-theoretic problems. Baker's work extended and made effective earlier results by mathematicians such as Alexander Gelfond and Theodor Schneider, and his methods found applications across several branches of mathematics. He remained associated with Cambridge throughout his career as a Fellow of Trinity College, contributing to the training of numerous doctoral students and publishing influential texts in his field.

Early life

Alan Baker was born on 19 August 1939 in London, England. He grew up during a period marked by the Second World War and its aftermath, which shaped the experiences of many British children of his generation. He demonstrated mathematical ability from an early age, and his aptitude set him on an academic trajectory that would eventually lead him to the University of Cambridge, one of the leading centres for mathematical research in the world.

Education

Alan Baker attended University College London (UCL) before moving to the University of Cambridge for his graduate studies. At Cambridge, he came under the supervision of Harold Davenport, one of the foremost number theorists of the twentieth century, whose influence on Baker's mathematical development was considerable. Baker pursued research in number theory, a branch of pure mathematics concerned with the properties and relationships of integers and related structures. His doctoral work at Cambridge laid the groundwork for the theoretical advances that would define his career, particularly his investigations into transcendental numbers — numbers that are not roots of any non-zero polynomial equation with integer coefficients. Cambridge's rich tradition in number theory, dating back to figures such as G. H. Hardy and J. E. Littlewood, provided a fertile intellectual environment for Baker's early research.

Career

Early academic work and Baker's theorem

After completing his graduate studies, Baker joined the faculty at the University of Cambridge, where he would remain for the bulk of his professional life. He was elected a Fellow of Trinity College, Cambridge, one of the oldest and most distinguished colleges within the university, connecting him to a long tradition of mathematicians associated with the college, including Isaac Newton, Srinivasa Ramanujan, and G. H. Hardy. Baker held a professorship in pure mathematics and was one of the central figures in Cambridge's mathematical community for several decades.

His research focused on transcendental number theory, a field that had seen major developments in the first half of the twentieth century through the work of mathematicians including Alexander Gelfond and Theodor Schneider. In 1934, Gelfond and Schneider had independently proved Hilbert's seventh problem, establishing that certain classes of numbers were transcendental. However, their methods were largely non-effective — they proved the existence of transcendental numbers in certain settings without providing quantitative bounds or practical methods for determining the transcendence of specific numbers.

Baker's central achievement was to develop effective methods in the theory of transcendental numbers. His work on linear forms in logarithms of algebraic numbers provided explicit lower bounds for expressions of the form:

β₁ log α₁ + β₂ log α₂ + ... + βₙ log αₙ

where the αᵢ are algebraic numbers and the βᵢ are algebraic numbers. Baker proved that such linear forms, when non-zero, cannot be arbitrarily small — more precisely, he established lower bounds for their absolute values in terms of the heights and degrees of the algebraic numbers involved. This result, now known as Baker's theorem, generalized the Gelfond–Schneider theorem from the case of two logarithms to an arbitrary finite number.

The significance of Baker's theorem extended well beyond the theory of transcendental numbers itself. By providing effective lower bounds, Baker's methods gave mathematicians tools to solve, or at least bound the solutions of, a wide range of Diophantine equations — polynomial equations where integer solutions are sought. Before Baker's work, many such equations could be shown to have finitely many solutions, but no practical method existed for determining what those solutions were or how large they could be. Baker's effective bounds changed this situation, making it possible in many cases to reduce the problem to a finite computation.

Fields Medal (1970)

In recognition of his contributions, Alan Baker was awarded the Fields Medal at the International Congress of Mathematicians held in Nice, France, in 1970. The Fields Medal is given every four years to mathematicians under the age of forty who have made outstanding contributions to the field. Baker was 31 years old at the time of the award.

The Fields Medal citation recognised Baker specifically for his work on the effective solution of problems in number theory, particularly his generalization of the Gelfond–Schneider theorem and the applications of his results to Diophantine equations. Baker was one of four recipients of the Fields Medal that year, alongside Heisuke Hironaka, Sergei Novikov, and John G. Thompson, each recognised for contributions in different areas of mathematics.

The award cemented Baker's reputation as one of the leading number theorists of his generation. His work was notable not only for its depth but for the breadth of its applications, connecting transcendental number theory to problems in algebraic number theory, Diophantine approximation, and mathematical logic.

Applications to Diophantine equations

One of the most celebrated applications of Baker's methods was to the class of Thue equations, named after the Norwegian mathematician Axel Thue. A Thue equation is a Diophantine equation of the form:

f(x, y) = m

where f is an irreducible binary form of degree at least three with integer coefficients and m is a non-zero integer. Thue had proved in 1909 that such equations have only finitely many integer solutions, but his proof was non-effective — it did not provide any bound on the size of the solutions. Using his results on linear forms in logarithms, Baker was able to give explicit upper bounds for the solutions of Thue equations, transforming Thue's qualitative result into a quantitative one.

Baker's methods were also applied to other classical problems in number theory, including the determination of all imaginary quadratic fields with a given class number. The class number problem, which asks for a classification of imaginary quadratic fields according to their class number, had been a major open question in algebraic number theory. Baker's effective methods contributed to the resolution of the class number one problem, confirming that there are exactly nine imaginary quadratic fields with class number one — a result that had been conjectured by Carl Friedrich Gauss and partially addressed by Kurt Heegner and Harold Stark.

Later career and publications

Throughout the 1970s, 1980s, and beyond, Baker continued to refine and extend his results on linear forms in logarithms. He and his collaborators improved the lower bounds in Baker's theorem, making them sharper and more applicable to specific problems. These improvements had consequences for computational number theory, where explicit bounds are essential for algorithmic approaches to solving Diophantine equations.

Baker was also active as an author and expositor of mathematics. His book Transcendental Number Theory, first published by Cambridge University Press in 1975, became a standard reference in the field. The book provided a systematic account of the theory of transcendental numbers, including Baker's own contributions, and was noted for its clarity and rigour. It introduced many graduate students and researchers to the methods and results of the field and remained influential for decades after its initial publication. Baker followed this with A Concise Introduction to the Theory of Numbers, published by Cambridge University Press in 1984, which offered a broader introduction to number theory accessible to undergraduate readers.

In addition to his monographs, Baker published numerous research papers in leading mathematical journals. His work attracted a substantial number of doctoral students to Cambridge, many of whom went on to make their own contributions to number theory and related fields. Baker's influence as a mentor and teacher extended the impact of his research well beyond his own publications, and the community of researchers working on linear forms in logarithms remained active into the twenty-first century.

Academic positions and affiliations

Baker held a professorship in pure mathematics at the University of Cambridge and served as a Fellow of Trinity College throughout much of his career. He was elected a Fellow of the Royal Society (FRS), the United Kingdom's national academy of sciences, in recognition of his contributions to mathematics — one of the highest honours available to scientists and mathematicians in the United Kingdom and the Commonwealth. He also received the Adams Prize from the University of Cambridge, an award given for distinguished contributions to the mathematical sciences.

Personal life

Alan Baker maintained a private personal life, and limited information is available in published sources about his personal interests outside of mathematics. He was known among colleagues and students for his dedication to mathematical research and his willingness to engage with students and junior researchers. Baker spent the majority of his adult life in Cambridge, where he was closely associated with the university and its mathematical community.

Baker died on 4 February 2018 in Cambridge, England, at the age of 78. His death was noted by the international mathematical community, with tributes recognising his contributions to number theory and his role in training the next generation of mathematicians.

Recognition

Alan Baker's contributions to mathematics were recognised through several major honours over the course of his career. The most prominent was the Fields Medal, awarded in 1970, which recognised his work on transcendental number theory and its applications to Diophantine equations. In addition to the Fields Medal, Baker received the Adams Prize from the University of Cambridge and was elected a Fellow of the Royal Society, reflecting the esteem in which his work was held by the broader scientific community. He was also invited to give lectures at international mathematical conferences throughout his career, including the International Congress of Mathematicians. His book Transcendental Number Theory received favourable reviews and was translated into multiple languages, extending the reach of his mathematical ideas to researchers worldwide.

Legacy

Alan Baker's legacy in mathematics rests primarily on the theoretical and practical impact of his work on linear forms in logarithms of algebraic numbers. Baker's theorem and its successive refinements provided the mathematical community with a set of tools that transformed several areas of number theory from qualitative to quantitative disciplines. Before Baker, many results in Diophantine approximation and transcendental number theory were known to hold in principle but could not be applied to specific numerical problems. Baker's effective methods bridged this gap, enabling mathematicians to solve concrete problems that had previously been intractable.

The influence of Baker's work can be measured in part by the extensive literature that built upon his results. Numerous mathematicians extended, refined, and applied Baker's methods in the decades following his initial publications. The so-called Baker method became a standard technique in computational number theory and was incorporated into algorithms for solving various classes of Diophantine equations. These methods have been implemented in computer algebra systems and continue to be used in mathematical research.

Baker's role as a teacher and mentor at Cambridge also contributed to his lasting influence. His doctoral students and academic descendants carried forward his research programme, applying and extending his methods to new problems and settings. The recognition Baker received during his lifetime — including the Fields Medal, Fellowship of the Royal Society, and the Adams Prize — reflected the significance of his contributions to mathematics. His books Transcendental Number Theory and A Concise Introduction to the Theory of Numbers continued to be cited and used as references decades after their initial publication, and his results remain part of the standard curriculum in advanced number theory courses at universities around the world.

Alan Baker's career exemplified the power of pure mathematical research to generate results of both theoretical depth and practical applicability. His work on transcendental numbers and Diophantine equations remains a cornerstone of modern number theory.

Other notable individuals named Alan Baker

The name Alan Baker is shared by several other notable individuals across different fields. Alan Baker (born 1956) is an American politician. Alan T. Baker (born 1956) is a United States Navy chaplain. Alan Baker (1944–2026) was an English footballer who played for Aston Villa. Alan Baker (born 1947) is a diplomat who served as Israel's Ambassador to Canada and has written extensively on legal and political matters relating to Israel and international law. Alan Baker (born 1938) is a British geographer. Alan Baker (born 1958) is a British poet. A further Alan Baker is a professor of philosophy also known as a shogi player. This article focuses on the mathematician Alan Baker (1939–2018), who is the most widely documented bearer of the name in encyclopaedic sources.