Definition:Hilbert 23/7
Hilbert $23$: Problem $7$
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".
Sources
- 1902: David Hilbert: Mathematical Problems (Bull. Amer. Math. Soc. Vol. 8, no. 10: pp. 437 – 479)
- (translated by Mary Winston Newson from "Mathematische Probleme")