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 work on transcendental number theory earned him the Fields Medal in 1970, one of mathematics' highest honours. Based mostly at the University of Cambridge, he developed what became known as Baker's theorem. This breakthrough transformed how mathematicians solve Diophantine equations and work with transcendental numbers. His results on linear forms in logarithms of algebraic numbers extended earlier work by Alexander Gelfond and Theodor Schneider, providing effective methods where they'd only given existence proofs. The impact rippled across several mathematical fields. Throughout his career, Baker stayed at Cambridge as a Fellow of Trinity College, mentored numerous PhD students, and published influential texts that shaped the discipline.

Early life

Baker was born on 19 August 1939 in London. Growing up during World War II and its aftermath shaped him like so many British children of that era. Mathematical talent showed early. By his teenage years, it was clear he'd found his calling in abstract mathematics. That aptitude pointed him toward Cambridge, then as now one of the world's leading centres for research in mathematics.

Education

He started at University College London (UCL) before moving to Cambridge for graduate work. There he studied under Harold Davenport, a towering figure in twentieth-century number theory. Davenport's influence shaped everything Baker would do next. His doctoral research focused on transcendental numbers, those strange values that aren't roots of any polynomial equation with integer coefficients. Cambridge's pedigree in this area mattered enormously. G. H. Hardy and J. E. Littlewood had built something remarkable there, and Baker benefited from that intellectual legacy.

Career

Early academic work and Baker's theorem

After finishing his PhD, Baker joined Cambridge's faculty. He became a Fellow of Trinity College, one of the university's oldest and most distinguished colleges. Isaac Newton, Srinivasa Ramanujan, and G. H. Hardy had all been there. Now Baker occupied a place in that tradition as a professor of pure mathematics and a central figure in Cambridge's mathematical life for decades.

Baker's research tackled transcendental number theory at a time when the field had recently seen major breakthroughs. Alexander Gelfond and Theodor Schneider had proved Hilbert's seventh problem in 1934, showing certain numbers were transcendental. But there was a crucial catch. Their proofs didn't provide actual numbers. They proved transcendental numbers existed without giving methods to find them or quantify their properties.

Baker changed this. He developed effective methods that actually worked on concrete problems. His breakthrough involved linear forms in logarithms of algebraic numbers, expressions like:

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

where the αᵢ are algebraic numbers and the βᵢ are algebraic numbers. Here's the key: Baker proved such forms cannot be vanishingly small. He established explicit lower bounds in terms of the heights and degrees involved. This result, now called Baker's theorem, extended the Gelfond–Schneider theorem from two logarithms to any finite number.

What made this so powerful extended far beyond transcendental number theory itself. These effective bounds gave mathematicians real tools for solving Diophantine equations. Before Baker, proving such equations had only finitely many solutions was possible. But actually finding those solutions or bounding their size? Nearly impossible. Baker's work made this practical. In many cases, his bounds reduced the problem to something a computer could handle.

Fields Medal (1970)

Baker received the Fields Medal in 1970 at the International Congress of Mathematicians in Nice, France. He was just 31 years old. The medal, awarded every four years to mathematicians under forty for exceptional work, represented the field's highest recognition for early-career achievement.

The citation specifically praised his work on effective solutions in number theory. His generalization of Gelfond–Schneider and applications to Diophantine equations were what caught the committee's attention. That year's four recipients were Baker, Heisuke Hironaka, Sergei Novikov, and John G. Thompson, each recognized for different areas.

The award cemented his status as a leading number theorist of his generation. What distinguished him wasn't just theoretical depth. His work connected multiple fields: transcendental number theory, algebraic number theory, Diophantine approximation, and mathematical logic.

Applications to Diophantine equations

Baker's methods found spectacular use in problems involving Thue equations. These are named after Axel Thue, a Norwegian mathematician. A Thue equation has 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 shown in 1909 these equations have finitely many solutions. His proof didn't work for practical purposes though. It proved solutions existed but gave no bounds on their size. Baker changed that. His results on linear forms in logarithms made Thue's qualitative finding quantitative and computable.

He also tackled the class number problem for imaginary quadratic fields. This classical problem asks which imaginary quadratic fields have a given class number. Baker's effective methods helped resolve the class number one problem, confirming exactly nine such fields exist. Gauss had conjectured this. Heegner and Stark worked on it. Baker's theorem made it rigorous and complete.

Later career and publications

Through the seventies, eighties, and beyond, Baker kept improving his bounds. He and collaborators sharpened them further, making them applicable to increasingly specific problems. For computational number theory, this mattered enormously. Explicit bounds are essential when algorithms need to solve Diophantine equations.

Baker was also a skilled expositor. His book Transcendental Number Theory, published by Cambridge University Press in 1975, became the standard reference. Graduate students learned the field from its pages. Researchers cited it for decades. The clarity was remarkable for such technical material. Next came A Concise Introduction to the Theory of Numbers in 1984, opening the subject to undergraduates.

Beyond these monographs, he published research steadily in top journals. His students went on to contribute to number theory and beyond. His mentoring extended his influence far beyond what publication alone could achieve. The entire community working on linear forms in logarithms traces back to his leadership.

Academic positions and affiliations

At Cambridge he held a professorship in pure mathematics and served as a Fellow of Trinity College through most of his career. The Royal Society elected him a Fellow (FRS), recognizing contributions at the highest level available to UK scientists and mathematicians. The University of Cambridge awarded him the Adams Prize for distinguished work in the mathematical sciences.

Personal life

Baker kept his personal life private. Published sources say little about interests outside mathematics. Colleagues and students knew him as dedicated to research and generous with his time. Cambridge was his home for most of his adult life, and the university's mathematical community was his world.

He died on 4 February 2018 in Cambridge at age 78. The international mathematical community marked his passing with tributes acknowledging both his theoretical contributions and his role in training the next generation.

Recognition

Major honours accumulated across his career. The Fields Medal in 1970 stood foremost, recognizing his transcendental number theory work and its Diophantine applications. Beyond that came the Adams Prize and election to the Royal Society, showing how widely his work was valued. International mathematical conferences invited him as a speaker throughout his life. Transcendental Number Theory was translated into multiple languages, spreading his mathematical ideas worldwide.

Legacy

Baker's mark on mathematics comes chiefly from his work on linear forms in logarithms. Baker's theorem and its refinements gave the mathematical community tools that changed several fields. What had been qualitative became quantitative. Before Baker, mathematicians knew many results should hold. They just couldn't apply them to actual problems. He bridged that gap.

The literature built on his results is extensive. Mathematicians extended, refined, and applied the Baker method for decades. It became standard in computational number theory. Computer algebra systems incorporated it. Research still relies on it today.

His mentoring at Cambridge amplified his legacy further. His doctoral students and their academic descendants carried the research programme forward. The honours he received during his lifetime—Fields Medal, Royal Society fellowship, Adams Prize—reflected the importance of his contributions. His two books continued to be cited long after publication and remain part of advanced number theory curricula worldwide.

Baker's career showed what pure mathematics could accomplish. Theoretical depth and practical applicability proved inseparable in his hands. His work on transcendental numbers and Diophantine equations remains fundamental to modern number theory.

Other notable individuals named Alan Baker

Several other notable people share this name across different fields. Alan Baker (born 1956) is an American politician. Alan T. Baker (born 1956) serves as a United States Navy chaplain. Alan Baker (1944–2026) was an English footballer with Aston Villa. Alan Baker (born 1947) is a diplomat who was Israel's Ambassador to Canada and has written extensively about legal and political issues involving Israel and international law. Alan Baker (born 1938) is a British geographer. Alan Baker (born 1958) is a British poet. There's also a philosopher and shogi player named Alan Baker. This article concerns the mathematician Alan Baker (1939–2018), who remains the most documented of these individuals in encyclopaedic sources.