Gelfond’s theorem

mathematics

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June 29, 2026

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Gelfond's theorem, proven by Aleksandr Gelfond in 1934, states that if a is an algebraic number (not 0 or 1) and b is an irrational algebraic number, then a^b is a transcendental number. A transcendental number is one that cannot be a solution to an algebraic equation with rational coefficients. This theorem solved one of the 23 famous problems posed by David Hilbert in 1900. Gelfond's work significantly advanced the study of transcendental numbers.12

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Aleksandr Osipovich Gelfond
This statement, now known as Gelfond’s theorem, solved the seventh of 23 famous problems that had been posed by the German mathematician David Hilbert in 1900. Gelfond’s methods were readily accepted by other mathematicians, and important new concepts in transcendental number theory were rapidly developed. Much of his work, including…
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This statement, now known as Gelfond’s theorem, solved the seventh of 23 famous problems that had been posed by the German mathematician David Hilbert in 1900. Gelfond’s methods were readily accepted by other mathematicians, and important new concepts in transcendental number theory were rapidly developed. Much of his work, including…
Read More