Chern–Simons theory: Difference between revisions

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'''Hilbert's seventh problem''' is one of [[David Hilbert]]'s [[Hilbert problems|list of open mathematical problems]] posed in 1900. It concerns the [[irrational number|irrationality]] and [[transcendental number|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 number|algebraic]] but [[irrational number|not rational]], is then the ratio between base and side always [[transcendental number|transcendental]]?
#Is <math>a^b</math> always [[transcendental number|transcendental]], for [[algebraic number|algebraic]] <math>a \not\in \{0,1\}</math> and [[irrational number|irrational]] algebraic <math>b</math>?


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 <math>a^b</math> is algebraic for algebraic ''a'' and rational ''b''.)
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From the point of view of generalisations, this is the case
 
:<math>b \ln{\alpha} + \ln{\beta} = 0</math>
 
of the general [[linear form in logarithms]] which was attacked by
Gelfond and then solved by [[Alan Baker (mathematician)|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==
*[[Hilbert number]]
 
==References==
* {{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 }}
 
== External links ==
* [http://aleph0.clarku.edu/~djoyce/hilbert/problems.html#prob7 English translation of Hilbert's original address]
 
{{Geometry-stub}}
 
{{Hilbert's problems}}
 
[[Category:Hilbert's problems|#07]]

Latest revision as of 03:15, 8 December 2014


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