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| | ==See also== |
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| | *[[Hilbert number]] |
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| | * {{cite book | editor=Felix E. Browder | editor-link=Felix Browder | title=Mathematical Developments Arising from Hilbert Problems | series=[[Proceedings of Symposia in Pure Mathematics]] | volume=XXVIII.1 | year=1976 | publisher=[[American Mathematical Society]] | isbn=0-8218-1428-1 | first=Robert | last=Tijdeman | authorlink=Robert Tijdeman | chapter=On the Gel'fond–Baker method and its applications | pages=241–268 | zbl=0341.10026 }} |
| | * {{cite book | first1=Yu. I. | last1=Manin | authorlink1=Yuri I. Manin | first2=A. A. | last2=Panchishkin | title=Introduction to Modern Number Theory | series=Encyclopaedia of Mathematical Sciences | volume=49 | edition=Second | year=2007 | isbn=978-3-540-20364-3 | issn=0938-0396 | zbl=1079.11002 | page=61 }} |
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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
always transcendental, for algebraic 
and irrational algebraic 
?
The second question was answered in the affirmative by Aleksandr Gelfond in 1934, and refined by Theodor Schneider in 1935. This result is known as Gelfond's theorem or the Gelfond–Schneider theorem. (The restriction to irrational b is important, since it is easy to see that is algebraic for algebraic a and rational b.)
From the point of view of generalisations, this is the case
of the general linear form in logarithms which was attacked by
Gelfond and then solved by Alan Baker.
It is called the Gelfond conjucture or Baker's theorem. Baker was rewarded
as a Fields Medal winner in 1970 because of this.
The first question is a consequence of the second question.
See also
References
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