Worked solution: Gelfond–Schneider transcendence proof via auxiliary functions (1934)

Step 1 of 7: Hilbert's seventh problem: is

2^{\sqrt{2}}

transcendental?

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In plain words

An algebraic number is one that solves some polynomial equation with whole-number coefficients, like $\sqrt{2}$ (root of $x^2-2$ ); a transcendental number, like $\pi$ or $e$ , solves no such equation. When David Hilbert listed his 23 problems in 1900, the seventh asked about numbers like $2^{\sqrt{2}}$ : you take an algebraic base $\alpha$ (here $2$ ) that is neither $0$ nor $1$ , raise it to an algebraic but irrational power $\beta$ (here $\sqrt{2}$ ), and ask whether the result must be transcendental.

Hilbert believed this was fantastically hard — he expected it to be solved only after the Riemann Hypothesis — yet Aleksandr Gelfond and Theodor Schneider each found a complete proof in 1934, independently, using a clever construction now called an auxiliary function.

\alpha^\beta \ \text{transcendental for algebraic } \alpha\notin\{0,1\},\ \beta \ \text{algebraic irrational}

Detailed analysis

In his 1900 address Mathematische Probleme, Hilbert posed as his seventh problem the question of whether $\alpha^\beta$ is transcendental whenever $\alpha$ is algebraic with $\alpha\ne0,1$ and $\beta$ is algebraic but irrational, taking $\alpha^\beta$ to mean any fixed value of $e^{\beta\log\alpha}$ for a choice of logarithm. Hilbert mentioned $2^{\sqrt{2}}$ and $e^\pi=(-1)^{-i}$ as concrete test cases.

The question sits at the meeting point of algebra (which numbers satisfy polynomial equations) and analysis (properties of the exponential function), and for over three decades it resisted every attempt. Gelfond proved the special case $e^\pi$ transcendental in 1929, then in 1934 both Gelfond and Schneider — working independently and with somewhat different auxiliary functions — proved the full statement, now called the Gelfond–Schneider theorem.

The proof strategy, standard in transcendence theory ever since, is proof by contradiction: assume $\gamma=\alpha^\beta$ is algebraic, build an auxiliary function using $\alpha,\beta,\gamma$ that is forced to be extremely small at many points, and then show this contradicts a general fact that nonzero algebraic numbers cannot be arbitrarily small. The remaining steps follow this route in the style presented by Yum-Tong Siu (Harvard, Math 113 lecture notes) and Serge Lang's classical textbook treatment.

Terms in this step

Algebraic and transcendental numbers: A complex number is algebraic if it is a root of some nonzero polynomial with integer (equivalently, rational) coefficients; otherwise it is transcendental. Every rational number and every root like $\sqrt{2}$ is algebraic; $\pi$ and $e$ are known to be transcendental.