Alan Baker wasn’t just a theorist. He was the guy who turned abstract number theory into a practical tool. Born in London on August 19, 1939, he died in Cambridge on February 4, 2018. But his legacy is etched in stone. Or rather, in equations.
He earned his B.S. from University College London in 1961. Then he moved to Trinity College, Cambridge. He got his M.A. and Ph.D. there by 1964. He stayed in academia, holding a post at University College for a year before joining Trinity’s faculty in 1966.
The real magic happened in 1970.
That’s when he received the Fields Medal at the International Congress of Mathematicians in Nice, France. It’s the Nobel Prize of mathematics. And Baker won it for work that changed how we understand Diophantine equations.
Solving the Unsolved Diophantine Equation
Before Baker, mathematicians knew there were limits. But they couldn’t prove them. Enter Baker.
He built on the work of giants. Axel Thue. Carl Ludwig Siegel. Klaus Friedrich Roth. These men laid the groundwork. Baker finished it.
His breakthrough? He proved that for a large class of equations, you can explicitly determine all solutions.
Let’s look at the math. If you have a Diophantine equation f(x, y) = m, where m is a positive integer and f is an irreducible binary form of degree n ≥ 3, there is an effective bound.
Call it B.
This bound depends only on n and the coefficients. For any solution (x0, y0 ), the maximum absolute value of x0 or y0 is less than or equal to B.
“His achievement was made all the more impressive by the German David Hilbert’s prediction…”
This isn’t just theory. It’s a bound. It’s a limit. It’s a way to stop guessing and start calculating.
Transcendental Number Theory and Gelfond-Schneider
But Baker didn’t stop at Diophantine equations.
He generalized the Gelfond-Schneider theorem. This theorem solved Hilbert’s seventh problem.
Here’s the gist: If α and β are algebraic, α ≠ 0, 1, and β is irrational, then α^β is transcendental.
Transcendental numbers. They don’t solve any algebraic equation. Pi is one. Euler’s number e is one.
Baker took this further. He showed that if α1, …, αk are algebraic (not 0 or 1), and if 1, β1, …, βk are linearly independent over the rationals, with all βi being irrational algebraic numbers, then the product α1^β1…αk^βk is also transcendental.
Why It Matters
Hungarian mathematician Paul Turán put it best at the Nice Congress.
“…his achievement was made all the more impressive by the German David
