About: Baker's theorem

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In transcendental number theory, a mathematical discipline, Baker's theorem gives a lower bound for the absolute value of linear combinations of logarithms of algebraic numbers. The result, proved by Alan Baker , subsumed many earlier results in transcendental number theory and solved a problem posed by Alexander Gelfond nearly fifteen years earlier.Baker used this to prove the transcendence of many numbers, to derive effective bounds for the solutions of some Diophantine equations, and to solve the class number problem of finding all imaginary quadratic fields with class number 1.

Attributes	Values
rdf:type	yago:WikicatMathematicalTheorems yago:WikicatTheoremsInAlgebra yago:WikicatTheoremsInComplexAnalysis yago:WikicatTheoremsInNumberTheory yago:WikicatTranscendentalNumbers yago:Abstraction100002137 yago:Communication100033020 yago:ComplexNumber113729428 yago:DefiniteQuantity113576101 yago:IrrationalNumber113730584 yago:Measure100033615 yago:Message106598915 yago:Number113582013 yago:Proposition106750804 yago:RealNumber113729902 yago:Statement106722453 yago:Theorem106752293 yago:TranscendentalNumber113730756
rdfs:label	مبرهنة باكر (ar) Baker's theorem (en) Théorème de Baker (fr) ベイカーの定理 (ja) Stelling van Baker (nl)
rdfs:comment	تدخل مبرهنة باكر في إطار نظرية الأعداد المتسامية، ونعطي أدنى حد لقيمة مطلقة من التركيبات الخطية من لوغاريتمات الأعداد الجبرية. سميت هذه المبرهنة هكذا نسبة إلى آلان باكر. (ar) In transcendental number theory, a mathematical discipline, Baker's theorem gives a lower bound for the absolute value of linear combinations of logarithms of algebraic numbers. The result, proved by Alan Baker , subsumed many earlier results in transcendental number theory and solved a problem posed by Alexander Gelfond nearly fifteen years earlier.Baker used this to prove the transcendence of many numbers, to derive effective bounds for the solutions of some Diophantine equations, and to solve the class number problem of finding all imaginary quadratic fields with class number 1. (en) Le théorème de Baker résout la conjecture de Gelfond. Publié par Alan Baker en 1966 et 1967, c'est un résultat de transcendance sur les logarithmes de nombres algébriques, qui généralise à la fois le théorème d'Hermite-Lindemann (1882) et le théorème de Gelfond-Schneider (1934). Ce théorème a été adapté au cas des nombres p-adiques par ; le permet de démontrer la conjecture de Leopoldt dans le cas d'un corps de nombres abélien, suivant un article d'Ax. (fr) ベイカーの定理（ベイカーのていり、英: Baker's theorem）とは、1966年-1968年にかけて、アラン・ベイカーによって発表された、対数関数の一次形式に対する線形独立性、および下界の評価に関する一連の定理のことである。下界の評価が計算可能であることから、数論の様々な分野で応用されている。 (ja) In de transcendentietheorie, een deelgebied van de wiskunde, geeft de stelling van Baker een ondergrens voor lineaire combinaties van logaritmen van algebraïsche getallen. De stelling is bewezen door en vernoemd naar de Britse wiskundige Alan Baker. De stelling van Baker verzamelde vele eerdere resultaten in de transcendentale getaltheorie en loste een probleem op dat bijna vijftien jaar eerder door Aleksander Gelfond was gesteld. Baker gebruikte dit om de transcendentie van veel getallen te bewijzen, om doeltreffende grenzen af te leiden voor de oplossingen van sommige diofantische vergelijkingen en om het probleem op te lossen van het vinden van alle imaginaire kwadratische velden met 1. (nl)
name	Baker's Theorem (en)
dcterms:subject	Theorems in number theory Transcendental numbers
Wikipage page ID	28751817 (xsd:integer)
Wikipage revision ID	1122130056 (xsd:integer)
Link from a Wikipage to another Wikipage	Cambridge University Press Quadratic field P-adic exponential function Algebraic independence Schanuel's conjecture Theorems in number theory Complex number Analytic subgroup theorem Transcendental numbers Gelfond–Schneider theorem Siegel's lemma Crelle's Journal Logarithm Six exponentials theorem Leopoldt's conjecture Mathematika Gisbert Wüstholz Alexander Gelfond Algebraic number Algebraic number theory Fields medal Number field Four exponentials conjecture Height function Academic Press Effective results in number theory Homogeneous polynomial Diophantine equations Dover Publications Auxiliary function Class number problem Rational number Vandermonde matrix Theodor Schneider Transcendental number theory Vandermonde determinant Springer-Verlag P-adic Class numbers of imaginary quadratic fields Hermite–Lindemann theorem
Link from a Wikipage to an external page	http://www.numdam.org/item%3Fid=SB_1969-1970__12__73_0 https://books.google.com/books%3Fisbn=0486495264 https://books.google.com/books%3Fisbn=3-540-66785-7 https://books.google.com/books%3Fisbn=052139791X https://books.google.com/books%3Fisbn=978-0-521-88268-2
sameAs	Baker's theorem Baker's theorem Baker's theorem Baker's theorem Baker's theorem Baker's theorem Baker's theorem Baker's theorem Baker's theorem Baker's theorem
dbp:wikiPageUsesTemplate	dbt:Citation dbt:Harvtxt dbt:Quotation dbt:Reflist dbt:Short_description dbt:Harvs dbt:Math_theorem
authorlink	Alan Baker (en)
first	Alan (en)
last	Baker (en)
year	1966 (xsd:integer) 1967 (xsd:integer)

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