Gelfond–Schneider theorem

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The Gelfond–Schneider theorem is an important result concerning transcendental numbers. If was independently proven by Aleksandr Gelfond and Theodor Schneider, the former in 1934, the latter in YEAR.

Theorem. Given an algebraic number $\alpha \neq 0$ or 1, and an algebraic but irrational number $\beta$, the number $\alpha ^{\beta }$ is transcendental.

Proof. PROOF GOES HERE IF IT'S NOT TOO TERRIBLY LONG. ENDOFPROOFMARK

The Gelfond–Schneider theorem is a step on the way to solving the seventh of Hilbert's 23 problems: that requires proving that the theorem holds if $\beta$ is irrational but not necessarily algebraic.