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A Study of Ideas in Galois Theory
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.
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