AI 中文总结
研究伽罗瓦《回忆录》中命题二，重构其核心内容，用现代术语刻画“置换群”和“代换群”，证明伽罗瓦群的代换是分裂域自同构，建立伽罗瓦原始表述与现代伽罗瓦理论表述的联系。
AI 中文摘要
在其手稿的命题二中，伽罗瓦写道：此证明有需完善之处。我没时间了。虽伽罗瓦未完成证明，但可重构命题二的核心内容并补充缺失论证。通常，V 表示线性形式 ax1+bx2+cx3+…，x1,x2,x3,…是可分多项式的不同根。设 L 是特征零的基域上的相应分裂域。一方面，命题二涉及 V 在包含于 L 的中间域 M 上的极小多项式 g(x)的因式分解；另一方面，涉及同一多项式分解为系数属于与 M 共轭的中间域的不可约因式。正是后一方面构成命题二的核心主题。“置换群”和“代换群”概念在《回忆录》中至关重要。我们用现代术语对这些概念进行了刻画。与每个“置换群”相关联的是一个多项式，其系数在相应“代换群”下不变。而且，一个“置换群”确定伽罗瓦群的一个划分，使得极小多项式 g(x)能分解为系数属于与相关“代换群”对应的中间域的因式。最后，我们证明伽罗瓦群的代换是分裂域 L 的域自同构。这建立了伽罗瓦原始表述与现代伽罗瓦理论表述之间的联系。
英文摘要
In Proposition II of his manuscript, Galois writes the well-known remark: Il y a quelque chose à compléter dans cette démonstration. Je n'ai pas le temps. Although Galois did not complete the proof, it is possible to reconstruct the essential content of Proposition II and to supply the missing arguments. As usual, V denotes the linear form ax1+bx2+cx3+..., where x1,x2,x3,... are distinct roots of a separable polynomial. Let L be the corresponding splitting field over a ground field of characteristic zero. On the one hand, Proposition II concerns the factorization of the minimal polynomial g(x) of V over an intermediate field M contained in L; on the other hand, it concerns the factorization of the same polynomial into irreducible factors whose coefficients belong to the intermediate fields conjugate to M. It is precisely this latter aspect that constitutes the central theme of Proposition II. The notions of 'groupe de permutations' and 'groupe de substitutions' are of fundamental importance in the Mémoire. We provide a characterization of these notions in modern terminology. Associated with every 'groupe de permutations' is a polynomial whose coefficients are invariant under the corresponding 'groupe de substitutions'. Moreover, a 'groupe de permutations' determines a partition of the Galois group, making it possible to factor the minimal polynomial g(x) into factors whose coefficients belong to the intermediate fields corresponding to the associated 'groupes de substitutions'. Finally, we prove that the substitutions of the Galois group are field automorphisms of the splitting field L. This establishes the connection between Galois' original formulation and the modern formulation of Galois theory.
