Hilbert's seventh problem
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Hilbert's seventh problem concerns the irrationality and transcendence of certain numbers (Irrationalität und Transzendenz bestimmter Zahlen). In its geometric formulation, it asks whether the following statement is provably true:
- In an isosceles triangle, if the ratio of the base angle to the angle at the vertex is algebraic but not rational, then the ratio between base and side is always transcendental.
A special case of this problem asks:
- Is ab transcendental, for algebraic a ≠ 0,1 and irrational algebraic b?
When b is rational, ab will be algebraic.
The special problem was solved by Aleksandr Gelfond in 1934, and refined by Theodor Schneider (1911 - ) in 1935. They proved that ab is transcendental when b is both algebraic and irrational. This result is known as Gelfond's theorem or the Gelfond-Schneider theorem.
From the point of view of generalisations, this is the case
- blog (α) + log(β) = 0
of the general linear form in logarithms.
See also: