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

Frequently asked questions

Why does the algebraic bound decay only like

e^{-CL}

while the analytic one decays like

e^{-cL\log L}

— where does the extra

\log L

come from?

The extra $\log L$ on the analytic side comes from Jensen's formula seeing $m$ separate zeros of order $L$ each — roughly $mL$ vanishing conditions, contributing a term with $\log r=\tfrac12\log L$ built in — whereas the algebraic (conjugate-counting) side only ever sees the single point $z_0=0$ and grows merely linearly in $L$ through the sizes of the coefficients involved. This mismatch in growth rate is exactly what the whole construction is designed to exploit.

Source

Aleksandr Gelfond and Theodor Schneider (independently), 1934

Sur le septième problème de Hilbert

Step 6 of 7: The algebraic lower bound: nonzero algebraic numbers can't be too small

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In plain words

Here is a simple but powerful fact: a nonzero rational number $p/q$ in lowest terms satisfies $|p/q|\ge1/q$ — it cannot be arbitrarily close to zero unless its denominator is huge. Algebraic numbers obey an analogous, slightly more elaborate rule: a nonzero algebraic integer's absolute value, multiplied together with the absolute values of all its 'conjugates' (the other roots of its minimal polynomial), must be at least $1$ in absolute value, because that product is always a nonzero ordinary integer.

Our number $\Phi^{(s)}(0)$ is (up to clearing an explicit, controllable denominator) exactly such an algebraic integer, built from $\alpha,\beta,\gamma$ and the coefficients of $G$ . Bounding the sizes of its conjugates using the same growth estimates as before shows it cannot be smaller than $e^{-CL}$ for an explicit constant $C$ — much larger (in absolute value) than the upper bound $e^{-cL\log L}$ from the previous step, once $L$ is large.

|\Phi^{(s)}(0)| \ge \exp(-C\, L)

Detailed analysis

For $\eta$ a nonzero algebraic number in a field $K$ with $[K:\mathbb{Q}]=D$ and 'denominator' $d$ (a positive integer clearing $\eta$ to an algebraic integer $d\eta$ ), the product of $|\sigma(d\eta)|$ over all embeddings $\sigma:K\hookrightarrow\mathbb{C}$ is a nonzero ordinary integer, hence has absolute value $\ge1$ ; isolating the one embedding giving $\eta$ itself yields $|\eta|\ge d^{-1}\prod_{\sigma\ne\mathrm{id}}|\sigma(\eta)|^{-1}$ — the algebraic analogue of $|p/q|\ge1/q$ (Siu, discussion of 'size'; this is standard in transcendence theory, e.g. Lang's Introduction to Transcendental Numbers).

Here $\eta=\Phi^{(s)}(0)/s!$ , an explicit polynomial expression in $\alpha,\beta,\gamma\in K$ and the coefficients of $G$ with denominator bounded using the same derivative-size estimates from Step 3 ( $\|D^\lambda(f^jg^k)\|\le(2J+L)^LC^{2J+L}$ , so denominators are of size $\lesssim L$ ). Bounding each conjugate $|\sigma(\eta)|$ requires estimating $\sigma$ applied to the whole construction — this uses the same growth bounds on derivatives, now applied uniformly over all embeddings of $K$ , giving $|\sigma(\eta)|\lesssim e^{C_1L}$ for an explicit $C_1$ depending on $[K:\mathbb{Q}]$ and the heights of $\alpha,\beta,\gamma$ .

Combining, $|\eta|=|\Phi^{(s)}(0)|/s!\ge e^{-C_2L}$ for an explicit $C_2$ (absorbing the denominator size and all conjugate bounds), so $\log|\Phi^{(s)}(0)|\ge-C_2L-\log(s!)\ge-C_3L\log L$ is too weak by itself, but comparing directly (without the $\log s!$ term, which the analytic side already accounted for) against Step 5's $\log|\Phi^{(s)}(0)|\lesssim-cL\log L$ shows the algebraic lower bound $-C_2L$ eventually exceeds the analytic upper bound $-cL\log L$ for $L$ large, since $L\log L$ grows strictly faster than $L$ — a genuine numerical contradiction.

Terms in this step

Height / size of an algebraic number: A numerical measure of the arithmetic complexity of an algebraic number $\eta$ , roughly the logarithm of the largest absolute value among $\eta$ 's denominator and all its Galois conjugates. Larger height means $\eta$ needs 'more room' to be written down exactly, and (dually) can be closer to zero without actually being zero.