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{{Distinguish|Gelfand–Naimark theorem}}
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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==
: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]]

Revision as of 12:23, 23 February 2014

With increasing property costs sky rocketing in Singapore and the Singapore government very eager to chill property prices to make it extra affordable to most people, many investors at the moment are taking a look at various locations for property investments.

The post A wave of recent property web sites is taking Singapore by storm appeared first on Vulcan Post CCS found that rents and costs of industrial properties in Singapore haven't been moving out of tandem, and that there was no proof of abnormal rents or price improve arising from anti-competitive practices. Properties in Singapore Bugis Junction Towers Marina Bay Financial Centre Section 1 Ocean Financial Centre One Raffles Quay Prudential Tower Buyer Stamp Responsibility on Buy of Properties – Commercial and residential properties b) Singapore Everlasting Residents (PR) who already own 1 or extra residential properties, whether or not owned wholly, partially, or jointly with others Try our function properties for a list of landed house for rent with photos and value.

Holders new launch sg of any diploma or greater tutorial qualification which has actual property as a serious are suggested to test with CEA whether they're required to take the REA/ RES examination. 2. Individuals who have passed all 3 examinations in Common Apply by the Royal Establishment of Chartered Surveyors (RICS) will likely be thought of as having obtained equivalent qualification to the REA examination. 3. Holders of skills in non-real estate related fields similar to enterprise administration, economics, advertising, law and many others will be required to take the REA/RES examination.

Since mid-2005 foreigners should buy flats (referred to as strata-titled properties ) in all buildings without having approval from the Singapore Authorities. Earlier rules in regards to the apartment block needing to be greater than six storeys and categorised as a condominium now not apply. Try our characteristic properties for a listing of landed house for sale with pictures and worth. Get to know the "property gross sales jargon" used in data offered by Singapore property brokers. Info Bankruptcy search After 5 years of excellent development with an immaculate observe file, Ken representing RE/MAX Singapore was provided the Grasp Franchise rights for Singapore for the following 40 years. EastLiving Group - Real Estate Agents - Real Property Singapore - 新加坡房地产公司 Property Tax TOP Date

Singapore is dedicated to deepening the manpower experience throughout the IP worth chain. The IP Academy of Singapore, established in 2002, gives skilled training across a wide space together with IP management, IP due diligence and transaction, and technology switch. The IP Academy additionally delivers graduate-degree programmes such as the Graduate Certificate in IP Legislation (GCIP) and the MSc in IP Management to boost the level of expertise of IP professionals in Singapore.

That is why for a residential sole company, we give recommendation on all features of the event - from image to design and structure, furnishings and facilities, and eventually venture advertising and sales. Our success rests on our extensive residential stock database to match shopper's needs with available properties; our skilled, local and regional expertise and wonderful market data, mixed with our consistency of service. The Panorama" in Ang Mo Kio by Hong Kong's Wheelock Properties was one of many top 5 main transactions by worldwide investors in 2013 - at S$550 million. By way of PropNex Worldwide, we regularly build on our reputation within the international property area. Click on here for extra of our abroad tasks.

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