Gelfond–Schneider theorem

For algebraic $a\ne 0,1$ and algebraic irrational $b$, every value $\exp (b\lambda )$ with $e^{\lambda }=a$ is transcendental. Consequently any quotient of chosen nonzero logarithms of algebraic numbers is either rational or transcendental.

Assume $a$ and $a^b$ algebraic with algebraic irrational $b$. Apply Siegel lemma to construct a small-coefficient polynomial $P$ vanishing at the algebraic pairs $(s+tb,a^s(a^b{)}^t)$ on a large square grid. The function $P(z,e^{\lambda z})$ is a nonzero exponential polynomial. Its many initial zeros make later grid values exponentially tiny, whereas an algebraic norm estimate forbids such tiny nonzero values. Iterating extends the zero grid indefinitely. Its quadratic zero count in disks contradicts the linear zero-count growth of an exponential polynomial.