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
historicReferences
History & 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.
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.