Gelfond-Schneider Theorem
The Gelfond-Schneider theorem states that if a and b are algebraic numbers, a is not zero or one, and b is irrational then ab is a transcendental number. For example, the theorem guarantees that 22 is transcendental by applying the result with a=2 and b = 2, which is irrational. It also guarantees that, for example, the base 10 logarithm of 2 is transcendental.
The statement of the Gelfond-Schneider theorem was posed as the seventh problem of D. Hilbert's famous list of 23 problems. Hilbert is reputed to have said that he did not expect this problem to be solved in his lifetime.
Contrary to his expectations, the theorem was proved independently by the Russian mathematician Aleksandr Osipovich Gelfond and the German mathematician Theodor Schneider in 1934. Their work is a vast extension of the work of C. Hermite and F. Lindemann who, in the 19th century, proved the transcendence of the numbers e and , respectively. The work of Gelfond and Schneider was further developed into modern transcendence theory by many authors, notably the British mathematician A. Baker who proved a conjecture of Gelfond on linear combinations with algebraic coefficients of logarithms of algebraic numbers, for which he received the Fields medal in 1966.
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