Hilbert's seventh problem
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Hilbert's seventh problem is one of David Hilbert's list of open mathematical problems posed in 1900. It concerns the irrationality and transcendence of certain numbers (Irrationalität und Transzendenz bestimmter Zahlen). Two specific questions are asked:
- In an isosceles triangle, if the ratio of the base angle to the angle at the vertex is algebraic but not rational, is then the ratio between base and side always transcendental?
- Is ab always transcendental, for algebraic
and irrational algebraic b?
The second question was answered in the affirmative by Aleksandr Gelfond in 1934, and refined by Theodor Schneider (1911–) in 1935. This result is known as Gelfond's theorem or the Gelfond–Schneider theorem. The answer is however unknown if b is allowed to be transcendental. (The restriction to irrational b is important, since it is easy to see that ab is algebraic for algebraic a and rational b.)
From the point of view of generalisations, this is the case
- blnα + lnβ = 0
of the general linear form in logarithms which was attacked by Alan Baker.
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