伽羅瓦第一論文之探討

Abstract

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。

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.