Gelfond–Schneider theorem

The theorem states that if α is algebraic and neither 0 nor 1, and β is algebraic but irrational, then every value of α^β is transcendental.

Algebraic number: A complex number that is a root of a nonzero polynomial with integer coefficients. The theorem’s base and exponent must both be algebraic numbers.
Linear forms in logarithms: Expressions formed as linear combinations of logarithms of algebraic numbers, studied through bounds and transcendence results. Bounds for such expressions underpin modern proofs of the theorem.
Alexander Gelfond: A Soviet mathematician whose work included a proof of the Gelfond–Schneider theorem. Gelfond independently established the theorem in 1934.
Gelfond's constant: The number e^π, which is transcendental by the Gelfond–Schneider theorem. It follows by taking the algebraic base −1 and exponent −i, with a suitable logarithm value.
Hermite's theorem: The theorem that e is transcendental, proved by Charles Hermite in 1873. It is an earlier, narrower transcendence result concerning one constant.
Transcendental number: A complex number that is not a root of any nonzero polynomial with integer coefficients. The theorem guarantees this status for every permitted power.
Hermite–Lindemann theorem: A theorem stating that the exponential of any nonzero algebraic number is transcendental. It is a foundational exponential-transcendence result used in the proof framework.
Theodor Schneider: A German mathematician known for work in number theory and transcendence. Schneider independently proved the same result in 1934.
2^√2: The real number obtained by raising 2 to the power √2; it is transcendental. It is a direct example with algebraic base and irrational algebraic exponent.
Liouville's theorem: A theorem giving rational-approximation criteria that produce transcendental numbers. It proves transcendence by approximation, not by algebraic exponentiation.

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