Galois理論思想研究

Abstract

十九世紀初，法國數學家Galois在他短暫的生命歷程中，留下代數學上極為珍貴的研究成果，並因著他獨創的理論思想，確立了他在數學史上劃時代的地位。

本論文將從西方數學的代數學發展脈絡中，認識Galois生長的時代背景與成長際遇，並深入探討Galois的數學理論，包括其思維的進路、理論的應用等，而對於Galois思想的特性以及文化上的意義也進行深入的研究。

全文共分五章。第一章「緒論」，說明研究動機、文獻探討、研究目的、研究方法等。第二章「Galois的生平背景」，以數學史的方法進行研究，對於代數學的前人足跡以及Galois在求學、研究及對政治狂熱的生平，詳細探討，並以年表呈現。第三章「Galois理論探討」，從數學的抽象代數角度剖析理論的層層關聯與整體輪廓，並加入許多例子輔以說明，其中不乏古希臘幾何三大難題與正n邊形作圖等問題，以作為理論的應用。第四章「Galois思想的文化意義」，用文化的眼光看待Galois思想，包括其理論特性與思想精神等，以及從數學美學的視野賞析Galois思想，並延伸至科學、藝術與音樂各領域，處處皆存有其思想的痕跡，最後回歸到數學教育的省思。第五章「結語」，以Galois從解決五次方程式根式解問題出發所發展出的理論思想卻能如此深邃豐沛作總結。

本論文研究的核心在第三章、第四章，期使偉大的Galois理論思想有新的詮釋。In early nineteenth century, the French mathematician Galois left his extremely valuable research results in algebra during the course of his short life. And because of the original theory and ideas which were developed on his own, he established a landmark in the history of mathematics.

This paper provides a view from the development context of algebra in Western mathematics. From knowing the historical background and growth of Galois to exploring the mathematical theory of Galois in depth, including his thinking approach, the application of the theory, and also study deeply in ideas and cultural meanings of Galois.

This paper is divided into five chapters. The first chapter, "Introduction," describes research motivation, purpose, and research methods. The second chapter, "Galois' life context", is the study using the history of mathematical methods to discuss in detail about the generation of predecessors footsteps in algebra, and the understanding of Galois’ schooling process, as well as his political fanaticism life. This chapter is chronologically presented.

The third chapter, "Discussion on Galois Theory", analyzes the association and the overall outline in the theory from the perspective of abstract algebra. It is supplemented by adding many examples to illustrate, including the three major problems in ancient Greek geometry, and the issue about weather the regular n-gon can be constructible by ruler and compasses, to apply the theory.

The fourth chapter, “The cultural significance of Galois’ thinking”, uses cultural vision to look at Galois’ ideas, including his theoretical characteristics and ideological spirit, as well as appreciating Galois’ thought from the perspective of mathematics aesthetic, and extends to various fields of science, art and music, which finally returns to the reflection of mathematics education. The fifth chapter, "Conclusion", summed the richness of Galois’ theoretical ideas, although he starts from solving the problem of the general equation of degree 5 by radicals.

The core of this thesis is in Chapter three and Chapter four, which expects the new interpretation in great ideas of Galois Theory.