%0 Journal Article
%T Non-Archimedean Analysis on the Extended Hyperreal Line *R<sub>d</sub> and the Solution of Some Very Old Transcendence Conjectures over the Field Q
%A Jaykov Foukzon
%J Advances in Pure Mathematics
%P 587-628
%@ 2160-0384
%D 2015
%I Scientific Research Publishing
%R 10.4236/apm.2015.510056
%X In 1980 F. Wattenberg constructed the Dedekind completion *R<sub>d</sub> of the Robinson non-archimedean field *R and established basic algebraic properties of *R<sub>d</sub>. In 1985 H. Gonshor established further fundamental properties of *R<sub>d</sub>. In [4] important construction of summation of countable sequence of Wattenberg numbers was proposed and corresponding basic properties of such summation were considered. In this paper the important applications of the Dedekind completion *R<sub>d</sub> in transcendental number theory were considered. Given any analytic function of one complex variable <img alt=\"\" src=\"Edit_894d76fb-61f7-4396-8faf-c3e450a0dcb9.jpg\" />, we investigate the arithmetic nature of the values of <img alt=\"\" src=\"Edit_c6ce20c1-245b-42aa-bfc8-62e95c94f723.jpg\" /> at transcendental points <img alt=\"\" src=\"Edit_e2b1b364-bf18-4d60-a7a0-27b1ab0351e8.jpg\" />. Main results are: 1) the both numbers <img alt=\"\" src=\"Edit_332a2b0e-2207-4797-b9d5-3ce2bee44faa.jpg\" /> and <img alt=\"\" src=\"Edit_2af59dea-857e-4f5f-a228-b1a0aef443c8.jpg\" /> are irrational; 2) number <em>e</em><sup><em>e</em></sup> is transcendental. Nontrivial generalization of the Lindemann-Weierstrass theorem is obtained.
%K Non-Archimedean Analysis
%K Robinson Non-Archimedian Field
%K Dedekind Completion
%K Dedekind Hyperreals
%K Wattenberg Embeding
%K Gonshor Idempotent Theory
%K Gonshor Transfer
%U http://www.scirp.org/journal/PaperInformation.aspx?PaperID=58868