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{{Distinguish|Gelfand–Naimark theorem}}


In [[mathematics]], the '''Gelfond–Schneider theorem''' establishes the [[transcendental number|transcendence]] of a large class of numbers. It was originally proved independently in 1934 by [[Aleksandr Gelfond]]<ref>{{cite journal |author=Aleksandr Gelfond |title=Sur le septième Problème de Hilbert |journal=Bulletin de l'Académie des Sciences de l'URSS. Classe des sciences mathématiques et na |volume=VII |issue=4 |pages=623–634 |year=1934 |url=http://mi.mathnet.ru/eng/izv4924}}</ref> and [[Theodor Schneider]]. The Gelfond–Schneider theorem answers affirmatively [[Hilbert's seventh problem]].


==Statement==
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:If ''a'' and ''b'' are [[algebraic number]]s with ''a''&nbsp;≠&nbsp;0,1 and if ''b'' is not a [[rational number]], then any value of ''a''<sup>''b''</sup> is a [[transcendental number]].
 
===Comments===
* The values of a and b are not restricted to [[real number]]s; [[complex number]]s are allowed (they are never rational when they have an imaginary part not equal to 0, even if both the real and imaginary parts are rational).
* In general, ''a''<sup>''b''</sup>&nbsp;=&nbsp;exp(''b''&nbsp;log&nbsp;''a'') is [[multivalued function|multivalued]], where "log" stands for the [[complex logarithm]]. This accounts for the phrase "any value of" in the theorem's statement.
* An equivalent formulation of the theorem is the following: if α and γ are nonzero algebraic numbers, and we take any non-zero logarithm of α, then (log&nbsp;γ)/(log&nbsp;α) is either rational or transcendental.  This may be expressed as saying that if log&nbsp;α, log&nbsp;γ are [[linear independence|linearly independent]] over the rationals, then they are linearly independent over the algebraic numbers. The generalisation of this statement to several logarithms of algebraic numbers is in the domain of [[transcendence theory]].
* If the restriction that ''a'' and ''b'' be algebraic is removed, the statement does not remain true in general. For example,
 
::<math>{\left(\sqrt{2}^{\sqrt{2}}\right)}^{\sqrt{2}} = \sqrt{2}^{\sqrt{2} \cdot \sqrt{2}} = \sqrt{2}^2 = 2.</math>
 
:Here, ''a'' is &radic;2<sup>&radic;2</sup>, which (as proven by the theorem itself) is transcendental rather than algebraic. Similarly, if ''a''&nbsp;=&nbsp;3 and ''b''&nbsp;=&nbsp;(log&nbsp;2)/(log&nbsp;3), which is transcendental, then ''a''<sup>''b''</sup>&nbsp;=&nbsp;2 is algebraic. A characterization of the values for ''a'' and ''b'', which yield a transcendental ''a''<sup>''b''</sup>, is not known.
* [[Kurt Mahler]] proved the [[P-adic number|''p''-adic]] analogue of the theorem: if ''a'' and ''b'' are in '''C'''<sub>''p''</sub>, the [[Complete metric space#Completion|completion]] of the [[algebraic closure]] of '''Q'''<sub>''p''</sub>, and they are algebraic over '''Q''', and if {{nowrap|{{!}}''a'' &minus; 1{{!}}<sub>''p''</sub> &lt; 1}} and {{nowrap|{{!}}''b'' &minus; 1{{!}}<sub>''p''</sub> &lt; 1}}, then (log<sub>''p''</sub>α)/(log<sub>''p''</sub>''b'') is either rational or transcendental, where log<sub>''p''</sub> is the [[P-adic exponential function#p-adic logarithm function|''p''-adic logarithm function]].
 
==Corollaries==
The transcendence of the following numbers follows immediately from the theorem:
 
* [[Gelfond–Schneider constant]] <math>2^{\sqrt{2}}</math> and its square root <math>\sqrt{2}^{\sqrt{2}}.</math>
* [[Gelfond's constant]] <math>e^{\pi} = \left( e^{i \pi} \right)^{-i} = (-1)^{-i} = 23.14069263 \ldots </math>, as well as <math> i^i = \left( e^{i \pi / 2} \right)^i = e^{-\pi / 2} = 0.207879576 \ldots.</math>
 
==See also==
* [[Lindemann–Weierstrass theorem]]
* [[Baker's theorem]]; an extension of the result
* [[Schanuel's conjecture]]; if proven it would imply both the Gelfond–Schneider theorem and the Lindemann–Weierstrass theorem
 
==References==
{{reflist}}
 
* {{Citation | last1=Baker | first1=Alan | author1-link=Alan Baker (mathematician) | title=Transcendental number theory | publisher=[[Cambridge University Press]] | isbn=978-0-521-20461-3 | year=1975 | zbl=0297.10013 | page=10}}
*{{Citation | last1=Feldman | first1=N. I. | last2=Nesterenko | first2=Yu. V. | authorlink2=Yuri Valentinovich Nesterenko | title=Transcendental numbers | publisher=[[Springer-Verlag]] | series=Encyclopedia of mathematical sciences | isbn=3-540-61467-2 | mr=1603604 | year=1998 | volume=44}}
*{{Citation | last1=Gel'fond | first1=A. O. | authorlink=Alexander Gelfond | title=Transcendental and algebraic numbers | origyear=1952 | url=http://books.google.com/books?isbn=0486495264 | publisher=[[Dover Publications]] | location=New York | series=Dover Phoenix editions | isbn=978-0-486-49526-2 | mr=0057921 | year=1960}}
*{{cite book | last = LeVeque | first = William J. | authorlink = William J. LeVeque | title = Topics in Number Theory, Volumes I and II | publisher = Dover Publications | location = New York | year = 2002 |origyear = 1956 | isbn = 978-0-486-42539-9 }}
* {{cite book | title=Irrational Numbers | first=Ivan | last=Niven | authorlink=Ivan M. Niven | publisher=Mathematical Association of America | year=1956 | isbn=0-88385-011-7 }}
* {{eom|id=G/g130020|first=Michel|last= Waldschmidt}}
* {{MathWorld|title=Gelfond-Schneider Theorem|urlname=GelfondsTheorem}}
 
==External links==
*[http://www.math.sc.edu/~filaseta/gradcourses/Math785/Math785Notes8.pdf A proof of the Gelfond–Schneider theorem]
 
{{DEFAULTSORT:Gelfond-Schneider theorem}}
[[Category:Transcendental numbers]]
[[Category:Theorems in number theory]]

Latest revision as of 18:53, 7 December 2014


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