《Galois theory》

H.E. p. 58 (S43)

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第七段

The main step in Galois' analysis of solvable equations is to...

---- 伽罗瓦对于可解方程的分析的主要步骤是...

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 ... study the way the group of an equation can be reduced by the adjunction of a pth root of a known quantity when the pth roots of unity are known.

---- 当第 p 个单位根已知时，通过添加已知量的第p个根，研究方程的群可被约减的方式。

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评论：“a (?) pth root of (?) a known quantity” 、“the pth roots (?) of unity are known” 含义待考。

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第六段

It is clear that if the field of K of known quantities is extended then the Galois group either remains the same or is reduced to a subgroup.

---- 清楚的是，如果域 K (已知量) 得到扩张，则伽罗瓦群：要么维持不变，要么约减到一个子群。

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This follows from the correspondence between the rows of the presentation of the Galois group and the roots t, t', t'', ... of the irreduible factor G(X) of F(X) of which t is a root;...

---- 这源于伽罗瓦群的表述阵列的诸行与F(x) 的不可约因式 G(X) 的诸根 t, t', t'', ... 之间的对应。

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... if K is enlarged then G(X) may no longer be irreducible, and the roots of the irreducible factor of F(X) of which t is a root may be a proper subset of t, t', t'', ..., in which case the new Galois group will be a proper subgroup of the old one.

---- 若 K 扩大则 G(X) 可能不再不可约，而 F(X) 的不可约因式 (t 是一个根) 的诸根可能是 t, t', t'', ... 的适当子集，这种情况下新的伽罗瓦群将是旧的伽罗瓦群的适当子群。

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Thus as the field increases K ⊂ K' ⊂ K'' ⊂... ⊂ K^(μ), the Galois group decreases.

---- 于是，当域扩大 K ⊂ K' ⊂ K'' ⊂... ⊂ K^(μ)，伽罗瓦群缩小。

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In the end, the roots a, b, c, ... have become “known” quantities -- that is, quantities in K^(μ) -- and, by Proposition 1, must all be left unchanged by all substitutions of the Galois group.

---- 最终，诸根 a, b, c, ... 成为 “已知的” 量 -- 即成为 K^(μ) 中的量 -- 而由命题1，它们全都在伽罗瓦群下不变。

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Thus, by the time K^(μ) is reached, the Galois group has been reduced to the identity substitution.

---- 这样，当达成 K^(μ) 时，伽罗瓦群已经缩减到单位置换。

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评论：这个第六段的概括很到位。

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