伽羅瓦第一論文之探討

A Survey of Galois' Premier Memoire

沈金緯(Jin-Wei Shen)

指導教授 ： 劉瓊如

國立臺灣大學/理學院/數學研究所/碩士(2014年)

https://doi.org/10.6342/NTU.2014.00949

摘要

本文回顧了Galois 最有名的論文，Premier Memoire。我們期望利用現代的
數學語言及工具走過Galois 及其前輩們所走過的路。其關鍵在於將Lagrange 的未定係數的方程式論延伸到一般係數的情形；並且在這個脈絡下引出體、(有限)群、Galois 預解形(Resolvent) 、Galois 群的定義。
市面上Galois 理論的相關出版物已汗牛充棟，而我們的目標在於定義Galois
群其的歷史源流及其構造的困難之處。而這一切都被年輕的Galois 完美解決了。

關鍵字

代數方程式的根式解 ； 群論 ； 體論 ； 自同構 ； 數學史

並列摘要

In this paper we study the original idea of the group of substitutions of an algebraic equation with either literal or numerical coefficients. The group of substitutions of the proposed roots or, equivalently, the group of automorphisms of the minimal splitting field containing the proposed roots, is the core of the theory of algebraic equation and is one of the motivations to the development of the modern
abstract group and the field theory. Although there are various publications of theory of algebraic equations and (finite) Galois theory. Our goal is the historical
background of the definition of a Galois group and the difficulties of explicit constructing them, which was solved by then young Galois.
The main obstacle of extending the theory from literal equations to arbitrary equations (literal or numerical) can be traced back to the work of Lagrange; and it is exactly Galois who solved the problem by his genius inventions of the Galois resolvent and the Galois group. In this paper we will inspect computation details of the algebraic solutions so that we can be fully motivated to see how subtle the
definition of a Galois group is made.

並列關鍵字

Radical solutions of algebraic equations ； group theory ； field theory ； automorphisms ； history of mathematics

參考文獻

[1] Artin, E. (1997). Galois Theory, Dover.

[2] Burnside, W. S. (2011). The Theory of Equations, Panton.

[3] Dickson, L. E. (1903). Introduction to The Theory of Algebraic Equa-

tions, John Wiley and Sons.

[4] Edwards, H. M. (1984). Galois Theory, Springer-Verlag.