Editing Gelfond–Schneider theorem

{{Use American English|date = January 2019}}
{{short description|On the transcendence of a large class of numbers}}
In [[mathematics]], the '''Gelfond–Schneider theorem''' establishes the [[transcendental number|transcendence]] of a large class of numbers.

==History==

It was originally proved independently in 1934 by [[Aleksandr Gelfond]] and [[Theodor Schneider]].<ref>{{Citation | last1 = Gelfond | first1 = A.| title = Sur le septième problème de D. Hilbert. (Russian, French) | url = | doi =  | journal = C. R. Acad. Sc. URSS | volume = 2 | issue = 2 | pages = 1–6 | year = 1934| JFM = 60.0163.04}}</ref><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 | JFM = 60.0164.01|url=http://mi.mathnet.ru/eng/izv4924}}</ref><ref>{{Citation | last1 = Gelfond | first1 = A.| title = On the seventh problem of D. Hilbert. (Sur le septième problème de D. Hilbert.) (Russian, French) | url = | doi =  | journal = C. R. (Dokl.) Acad. Sci. URSS  | volume = 2 | issue =| pages = 1–3, 4-6 | year = 1934| zbl = 0009.05302}}</ref><ref>{{Citation | last1 = Schneider | first1 = Theodor| title = Transzendenzuntersuchungen periodischer Funktionen. I. Transzendenz von Potenzen. | url = | doi =  | journal = Journal für die Reine und Angewandte Mathematik (Crelle)| volume = 172 | issue = | pages = 65–69 | year = 1934| zbl = 0010.10501}}</ref>

==Statement==

{{block indent|1=If ''a'' and ''b'' are [[algebraic number]]s with ''a'' <math>\not\in \{0,1\}</math> and ''b'' not [[Rational number|rational]], 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 (here complex numbers are not regarded as rational when they have an imaginary part not equal to 0, even if both the real and imaginary parts are rational).

In general, {{nowrap|1=''a<sup>b</sup>'' = exp(''b'' log ''a'')}} is [[multivalued function|multivalued]], where log stands for the complex [[natural logarithm]]. (This is the multivalued inverse of the exponential function exp.) 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 {{nowrap|(log ''γ'')/(log ''α'')}} is either rational or transcendental.  This may be expressed as saying that if {{nowrap|log ''α''}}, {{nowrap|log ''γ''}} are [[linear independence|linearly independent]] over the rationals, then they are linearly independent over the algebraic numbers.  The generalisation of this statement to more general [[linear forms in logarithms]] of several algebraic numbers is in the domain of [[transcendental number theory]].

If the restriction that ''a'' and ''b'' be algebraic is removed, the statement does not remain true in general. For example,
<math display="block">{\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 {{nowrap|1=''a'' = 3}} and {{nowrap|1=''b'' = (log 2)/(log 3)}}, which is transcendental, then {{nowrap|1=''a<sup>b</sup>'' = 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 <math>|a-1|_p<1</math> and <math>|b-1|_p<1,</math> then <math>(\log_p a)/(\log_p b)</math> 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}}=2.665144142\ldots </math> and its square root <math>\sqrt{2}^{\sqrt{2}}=\sqrt{2^{\sqrt{2}}}= 1.632526919\ldots</math>
* [[Gelfond's constant]] <math>e^{\pi} = \left( e^{i \pi} \right)^{-i} = (-1)^{-i} = 23.14069263 \ldots</math> 
* <math> i^i = \left( e^{\frac{i \pi}{2}} \right)^i = e^{-\frac{\pi}{2}} 
= \frac{1}{\sqrt{e^{\pi}}} = 0.207879576 \ldots</math>

==Applications==

The Gelfond–Schneider theorem answers affirmatively [[Hilbert's seventh problem]].
==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}}
===Further reading===
* {{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. | author-link2=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. | author-link=Alexander Gelfond | title=Transcendental and algebraic numbers | orig-year=1952 | url=https://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. | author-link = William J. LeVeque | title = Topics in Number Theory, Volumes I and II | publisher = Dover Publications | location = New York | year = 2002 | orig-year = 1956 | isbn = 978-0-486-42539-9 | url-access = registration | url = https://archive.org/details/topicsinnumberth0000leve }}
* {{cite book | title=Irrational Numbers | url=https://archive.org/details/irrationalnumber00nive | url-access=registration | first=Ivan | last=Niven | author-link=Ivan M. Niven | publisher=Mathematical Association of America | year=1956 | isbn=0-88385-011-7 }}

==External links==
* {{MathWorld|title=Gelfond-Schneider Theorem|urlname=GelfondsTheorem}}
*[https://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]]