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[注:下文是群邮件内容。]
《Galois theory》 H.E. p. 54 (S42) * * * 10:44 第一段 Galois gives two examples of Galois groups of equations. ---- 伽罗瓦给出了方程的伽罗瓦群的两个例子。 . The first is what he calls an “algebraic” equation, by which he means an equation in which the coefficients are indeterminates rather than numbers. ---- 第一个例子他称之为 “代数” 方程,指方程的系数是待定元 (而不是数值)。 . For such equations, he says that the group is the set of all n! substitutions of the roots “because in this case the symmetric functions [of the roots] are the only ones that can be determined rationally, ” that is, are the only ones in K. ---- 对这种方程,他指出群是根的所有 n! 个置换 “因为在此情况中 [根的] 对称函数是唯一能有理地确定的”,即唯一在 K 中 的。 . In other words, only symmetric functions of the roots can be expressed in terms of the coefficients and therefore, by Proposition I, the group must contain all substitutions. ---- 换句话说,只有根的对称函数可以由系数表达,因此 (由命题1) 群必须包含全部置换。 . This is rather plausible, but the proof is far from obvious. (See S67.) ---- 这相当合理,但其证明却绝非显然。(见S67)。 . 评论:伽罗瓦设想出 “代数” 方程,其预解群为根的全部置换。(注:我把伽罗瓦群也称作 “预解群”)。 . 小结:这一段只是略提及伽罗瓦给出的第一个例子(没有技术内容)。 * * *19:55 符号大全、上下标.|| 常用:↑↓ π ΓΔΛΘΩμφΣ∈ ∉ ∪ ∩ ⊆ ⊇ ⊂ ⊃ Ø ∀ ∃ ≤ ≥ ⌊ ⌋ ⌈ ⌉ ≠ ≡ ⁻⁰ 1 2 3 ᵈ ⁺ ₊ ₀ ₁ ₂ ₃ ᵢ . |
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