The problem
Theorem: if a and b are algebraic numbers with a ≠ 0, a ≠ 1, and b irrational, then any value of \(a^b\) = exp(b log a) is transcendental.
The Gelfond–Schneider Theorem
historicHistory & significance
Hilbert's seventh asked whether α^β is transcendental for algebraic α ∉ {0,1} and irrational algebraic β. He considered it harder than RH ('may have to await till the discovery of entirely new principles') and mentioned specific cases: \(2^{√2}\) and the Gelfond–Schneider constant e^π = (−1)^{−i}. Gelfond proved it in 1934 using interpolation determinants; Schneider independently found a different proof months later.
Connected problems
- inspired by Schanuel's Conjecture
- related Transcendence of Euler's Constant γ
More in transcendental number theory
- same area Schanuel's Conjecture
References
The resolution (human proof)
Independently proven by Alexander Gelfond and Theodor Schneider, 1934. The method extends Hermite–Lindemann transcendence via auxiliary polynomials vanishing to high order at conjugate points — the ancestor of Baker's theorem on linear forms in logarithms (1975 Fields Medal). Immediate consequences: e^π = (−1)^{−i} is transcendental; the area of the unit circle divided by the side of the square on its radius (π/√2 relation) is transcendental; and the Gelfond–Schneider constant is named for the theorem. Hilbert underestimated the difficulty: 'entirely new principles' took only thirty-four years.
How to check it: the skeleton is Siegel's lemma (small auxiliary polynomial) plus extrapolation to conjugates — abstracted later as the Schneider–Lang criterion. Baker's theory of linear forms in logarithms (1966 onward) generalises the method to many logarithms at once. See Schanuel's conjecture here for the statement that would subsume all of it.
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