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

Frequently asked questions

Does this proof give any information about how close

\alpha^\beta

is to being algebraic, or just that it isn't?

This particular qualitative argument only shows transcendence — it gives no explicit lower bound on how well $\alpha^\beta$ can be approximated by algebraic numbers. Later work by Baker and others (linear forms in logarithms, building on the same auxiliary-function philosophy) does give effective, quantitative versions of such statements, with real applications to Diophantine equations like Catalan's.

Source

Aleksandr Gelfond and Theodor Schneider (independently), 1934

Sur le septième problème de Hilbert

Step 7 of 7: The contradiction closes:

\alpha^\beta

must be transcendental

Previous step Next step

In plain words

Now the trap springs shut. Step 5 showed the number $\Phi^{(s)}(0)$ must be smaller (in absolute value) than roughly $e^{-cL\log L}$ . Step 6 showed the very same number, being a nonzero algebraic number, cannot be smaller than roughly $e^{-CL}$ . Since $L\log L$ grows faster than any fixed multiple of $L$ , for $L$ large enough these two requirements flatly contradict each other: the number would have to be both smaller than a rapidly shrinking bound and bigger than a more slowly shrinking one.

The only way out is that one of the assumptions was false — and everything else in the construction (Siegel's lemma, the growth estimates, the algebraic number theory) was airtight and used no unproven hypothesis. The only assumption left standing is the one made at the very start: that $\gamma=\alpha^\beta$ is algebraic. So that assumption must be wrong, and $\alpha^\beta$ is transcendental after all.

-cL\log L \ge \log|\Phi^{(s)}(0)| \ge -C L \implies \text{contradiction for } L \gg 1

Detailed analysis

Step 5 established $\log|\Phi^{(s)}(0)|\le-cL\log L+O(L)$ for an explicit constant $c>0$ ; Step 6 established $\log|\Phi^{(s)}(0)|\ge-C_2L$ for an explicit constant $C_2$ . Both bounds hold simultaneously (they are estimates on the very same nonzero number $\Phi^{(s)}(0)\in K$ ), so for all sufficiently large $L$ , $-C_2L\le\log|\Phi^{(s)}(0)|\le-cL\log L+O(L)$ , i.e. $-C_2L\le-cL\log L+O(L)$ . Dividing by $L$ and letting $L\to\infty$ , the right side tends to $-\infty$ while the left side stays bounded — an outright contradiction for $L$ larger than some explicit threshold depending only on $c,C_2,[K:\mathbb{Q}]$ (Siu, final paragraph of the proof of the Main Theorem, concluding ' $m\le C_7\rho[K:\mathbb{Q}]$ ', the analogous bound in the general statement).

Every step of the construction — Siegel's lemma (Step 3), the algebraic independence of $e^z,e^{\beta z}$ (Step 2), the growth estimates for entire functions of finite order (Steps 5–6), and the product-over-conjugates estimate for algebraic numbers (Step 6) — is unconditional classical mathematics. The only hypothesis introduced was the assumption in Step 2 that $\gamma=\alpha^\beta$ is algebraic. Since that hypothesis leads to a contradiction, it must be false: $\alpha^\beta$ is transcendental for every algebraic $\alpha\notin\{0,1\}$ and every algebraic irrational $\beta$ — this is the Gelfond–Schneider theorem, resolving Hilbert's seventh problem (Gelfond 1934; Schneider 1934, independently).

As an immediate corollary (Siu, 'Corollary 2'), Hilbert's example $2^{\sqrt{2}}$ is transcendental, since $\alpha=2$ is algebraic (not $0$ or $1$ ) and $\beta=\sqrt2$ is algebraic irrational. Similarly $\gamma=(-1)^{-i}=e^\pi$ is transcendental, recovering Gelfond's earlier 1929 special case, though establishing $e^\pi$ 's transcendence formally uses $\alpha=-1$ (a root of unity) and $\beta=i\sqrt{-1}$ -type reasoning with slightly adjusted hypotheses, both handled by the general Gelfond–Schneider machinery.