Definition:Hilbert 23/7

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Hilbert $23$: Problem $7$

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Gelfond-Schneider Theorem

Let $\alpha$ and $\beta$ be algebraic numbers (possibly complex) such that $\alpha \notin \set {0, 1}$.

Let $\beta$ be irrational.

Then any value of $\alpha^\beta$ is transcendental.

Historical Notes

Historical Note on Hilbert's $7$th Problem

David Hilbert believed that determining whether certain numbers are irrational or transcendental would be a problem more difficult than the Riemann Hypothesis or Fermat's Last Theorem.

Two specific cases, however, were resolved relatively quickly:

$e^\pi$ (Gelfond's constant) was proved transcendental in $1929$
$2^{\sqrt 2}$ (the Gelfond-Schneider constant) was proved transcendental in $1930$.

Subsequently they were seen as special cases of the Gelfond-Schneider Theorem, which was proved in $\text {1934}$ – $\text {1935}$.

It is sometimes pointed out that, in view of the fact that Fermat's Last Theorem held out till $1994$ and the Riemann Hypothesis still refuses to yield, in this case Hilbert's pessimism was unfounded.

However, note that certain other numbers of a similar type: it has still not been established whether $2^e$, $\pi^e$ and $2^\pi$, for example, are transcendental or not.

So, in general it may be accepted that Hilbert was not altogether incorrect.

Historical Note on Hilbert $23$

The Hilbert 23 were delivered by David Hilbert in a famous address at Paris in $1900$.

He considered them to be the oustanding challenges to mathematicians in the future.

There was originally going to be a $24$th problem, on a criterion for simplicity and general methods in proof theory, but Hilbert decided not to include it, as it was (like numbers $4$, $6$, $16$ and $23$) too vague to ever be described as "solved".


(translated by Mary Winston Newson from "Mathematische Probleme")