Editing Hilbert's seventh problem

{{Short description|On transcendence of certain numbers}}
'''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'').

==Statement of the problem==
Two specific equivalent<ref>{{cite book|first=N. I. |last=Feldman|authorlink=Naum Il'ich Feldman |first2= Yu. V. |last2=Nesterenko|editor-last1=Parshin|editor-first1=A. N.|editor-last2=Shafarevich|editor-first2=I. R.|series=Number Theory IV |title=Transcendental Numbers|url=https://archive.org/details/numbertheorytran00pars |url-access=limited |date=1998|publisher=Springer-Verlag Berlin Heidelberg|isbn=978-3-540-61467-8|pages=[https://archive.org/details/numbertheorytran00pars/page/n145 146]–147}}</ref> 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>?

==Solution==
The question (in the second form) 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''.)

From the point of view of generalizations, this is the case

:<math>b \ln{\alpha} + \ln{\beta} = 0</math>

of the general linear form in logarithms, which was studied by Gelfond and then solved by [[Alan Baker (mathematician)|Alan Baker]]. It is called the Gelfond conjecture or [[Baker's theorem]]. Baker was awarded a [[Fields Medal]] in 1970 for this achievement.

==See also==
*[[Gelfond–Schneider constant|Hilbert number ''or'' Gelfond–Schneider constant]]

==References==
{{Reflist}}

== Bibliography ==

* {{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=978-0-8218-1428-4 | 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]

{{Hilbert's problems}}
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[[Category:Hilbert's problems|#07]]