Linear form in logarithms
In transcendental number theory, a mathematical discipline, Baker's theorem gives a lower bound for the absolute value of linear combinations of logarithms of algebraic numbers. The result, proved by Alan Baker (1966, 1967a, 1967b), subsumed many earlier results in transcendental number theory and solved a problem posed by Alexander Gelfond nearly fifteen years earlier. Baker used this to prove the transcendence of many numbers, to derive effective bounds for the solutions of some Diophantine equations, and to solve the class number problem of finding all imaginary quadratic fields with class number 1.
To simplify notation, we introduce the set L of logarithms of nonzero algebraic numbers, that is
Using this notation, several results in transcendental number theory become much easier to state. For example the Hermite–Lindemann theorem becomes the statement that any nonzero element of L is transcendental.
In 1934, Alexander Gelfond and Theodor Schneider independently proved the Gelfond–Schneider theorem. This result is usually stated as: if a is algebraic and not equal to 0 or 1, and if b is algebraic and irrational, then ab is transcendental. Equivalently, though, it says that if λ1 and λ2 are elements of L that are linearly independent over the rational numbers, then they are linearly independent over the algebraic numbers. So if λ1 and λ2 are elements of L and λ2 isn't zero, then the quotient λ1/λ2 is either a rational number or transcendental. It can't be an algebraic irrational number like √2.
Although proving this result of "rational linear independence implies algebraic linear independence" for two elements of L was sufficient for his and Schneider's result, Gelfond felt that it was crucial to extend this result to arbitrarily many elements of L. Indeed, from Gel'fond (1960, p. 177):
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