Linearly Independent Sets and Transcendental Numbers

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题名:	Linearly Independent Sets and Transcendental Numbers
作者:	林宜萱;Lin, Yi-Hsuan
贡献者:	數學系
关键词:	線性獨立集;超越數
日期:	2020-07-24
上传时间:	2020-09-02 18:47:08 (UTC+8)
出版者:	國立中央大學
摘要:	在線性代數中我們知道任何佈於一體 (over a field) 的向量空間 (vector space) 存在至少一組基底。這篇碩士論文的初始動機起源於一個由呂明光教授提出的問題：對佈於有理數體 Q 上的向量空間實數體 R，是否存在一組確切的基底？對此目前我們沒有給出一個答案，但在網路上搜尋到 F. G. Dorais 提供了一組確切佈於 Q 上的線性獨立實數子集 T，並且 \|T\| = \|R\|。從集合 T 我們衍生出一些例子，同樣是佈於 Q 上的線性獨立實數子集，並且其集合大小與 \|R\| 相同。由於代數數有無窮可數多個而實數有無窮不可數多個，另一個由呂明光教授提出的問題是決定 Dorais 提供的集合 T 中哪些數是超越數〔即，非代數數〕。為了回答這個問題，我們研讀 Edward B. Burger 和 Robert Tubbs 的書 Making transcendence transparent. An intuitive approach to classical transcendental number theory。這個問題還沒被解決，然而 Burger 與 Tubbs 的書中介紹了 Liouville 數〔一種特別的超越數〕，在此我們導出一些例子作為練習。;It is known in linear algebra that every vector space over a field has a basis. The motivation of this thesis is to answer a question asked by Professor Ming-Guang Leu: Is there an explicit basis for the field R of real numbers over the field Q of rational numbers? To that we have yet no answer. However, it is found on the Internet that F. G. Dorais provides an explicit linearly independent subset T of R over Q with \|T\| = \|R\|. Inspired by the set T, we give some examples of linearly independent subsets of R over Q with the same cardinality as \|R\|. Since there are countably many algebraic numbers while there are uncountably many real numbers, another question asked by Professor Leu is to determine which number in the set T, given by Dorais, is a transcendental number (i.e., not an algebraic number). To answer the question, we study the book Making transcendence transparent. An intuitive approach to classical transcendental number theory by Edward B. Burger and Robert Tubbs. The question is not yet answered. However, in Burger and Tubbs′ book, Liouville numbers (a special type of transcendental numbers) are introduced, and we derive some examples of Liouville numbers as exercises.
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