# [学习笔记] H.E. p.61

[注：下文是群邮件的内容。]

《Galois theory》

H.E. p. 61 (S45)

* * *

The preceding two articles show that if f(x) = 0 is solvable by radicals then its Galois group G has a sequence of subgroups G ⊃ G' ⊃ G'' ⊃ ... ⊃ G^(ν) such that...

---- 前两篇文章表明，如果 f(x) = 0 是根式可解的，则它的伽罗瓦群 G 有子群序列 G ⊃ G' ⊃ G'' ⊃ ... ⊃ G^(ν) 使得...

.

.

...each group G^(i) is a normal subgroup of prime index in its predecessor G^(i-1), and such that the final subgroup G^(ν) consists of the identity substitution alone.

---- (使得) 每个群 G^(i) 是前一个群 G^(i-1) 的正规子群 (指标为素数)，并使得最后的子群 G^(ν) 只由单位置换组成。

.

.

Indeed, for this one needs only to take the sequence of field extensions K ⊂ K' ⊂ K'' ⊂... ⊂ K^(μ) implied by the solution by radicals, ...

---- 确实，为此你只须取由根式解蕴含的域扩张序列 K ⊂ K' ⊂ K'' ⊂... ⊂ K^(μ)，...

.

... to take the Galois groups of the equation f(x) = 0 reletive to each of the fields K^(i), and to disregard groups that coincide with their predecessors.

---- 以得到方程 f(x) = 0 关于每个域 K^(i) 的伽罗瓦群，并剔除与先前发生重复的群。

.

.

.

A group G is said to be solvable if it has such a sequence of subgroups.

---- 群 G 称作是可解的如果它有这样的一个子群序列。

.

Thus it has been shown that if an equation* f(x) = 0 is solvable by radicals then its Galois group is solvable.

---- 于是这就已经证明如果方程 f(x) = 0 是根式可解的，则它的伽罗瓦群是可解的。

.

Galois showed that this necessary condition for solvability by radicals is also sufficient.

---- 伽罗瓦证明根式可解的这个必要条件也是充分的。

.

More specifically, he showed that if G is the Galois group of f(x) = 0 over K and if G ⊃ G' ⊃ ... ⊃ G^(ν) is a sequence of of subgroups of G in which each G^(i) is a normal subgroup of prime index in its predecessor and G^(ν) = {identity}...

.

---- 更具体地，他证明了，如果 G 是 f(x) = 0 在 K 之上的伽罗瓦群，并且如果 G ⊃ G' ⊃ ... ⊃ G^(ν) 是 G 的一个子群序列，其中 G^(i) 是其前面子群的素指标正规子群，并且 G^(ν) = {单位置换} ...

.

...then there is a sequence of field extensions  K ⊂ K' ⊂ K'' ⊂... ⊂ K^(μ)  such that K^(μ) contains all the roots of f(x) = 0 and such that each extension K^(i-1)  K^(i) is obtained by adjunction of a pth root of a quantity of K^(i-1) for some prime p (depending on i).

---- 则存在一个域扩张序列 K ⊂ K' ⊂ K'' ⊂... ⊂ K^(μ) 使得 K^(μ) 包含 f(x) = 0 的所有的根，并且使得每个扩张 K^(i-1) ⊂ K^(i) 是由添加 K^(i-1)中的一个量的 p 次根而获得的，这里 p 是某个素数 (依赖于 i)。

.

.

The main element in the proof of this fact is the proof of a converse of the proposition of S44.

---- 这一事实的证明之主要元素乃是 S44 中命题的逆之证明。

.

This converse is the subject of the next article.

---- 这个逆命题是下一篇文章的主题。

.

.

* * *

http://blog.sciencenet.cn/blog-315774-1252452.html

## 全部精选博文导读

GMT+8, 2020-10-29 05:13