Baker's theorem

Baker's theorem (1966), proved by Alan Baker, states that if α₁, ..., αₙ are algebraic numbers not equal to 0 or 1, and β₁, ..., βₙ are algebraic and linearly independent over the rationals, then the product α₁^β₁ ... αₙ^βₙ is transcendental. This was the culmination of a line of work beginning with the Gelfond-Schneider theorem (1934), and it earned Baker the Fields Medal in 1970.

The deeper significance of Baker's work lies not in the specific transcendence result but in the method: Baker provided effective lower bounds for linear forms in logarithms of algebraic numbers. Where previous transcendence proofs showed that certain quantities were non-zero, Baker's bounds quantified how far from zero they were. This effectiveness transformed Diophantine approximation from a non-constructive finiteness theory into a toolkit for explicitly solving equations.

The applications were immediate and sweeping. Baker's methods gave the first effective bounds for Thue equations, for elliptic curve point counts, and for the Catalan conjecture (eventually proved by Mihăilescu). The theorem is the bridge between the abstract world of transcendence and the concrete world of computational number theory.