《Galois theory》

H.E. p. 57 (S43)

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The essence of Galois' achievement was to determine the conditions imposed on the Galois group of an equation by the assumption that the equation is solvable by radicals.

---- 伽罗瓦的本质成就乃是通过假定方程根式可解来确定施加给方程的伽罗瓦群上的条件。

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The first step in doing this, naturally, is to state very explicitly what it means to say that an equation is solvable by radicals.

---- 做这件事的第一步，自然地，乃是讲清楚方程根式可解究竟是什么意思。

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评论：既然每个多项式方程都有个 “预解群”，那么方程的一切性质都可以从中得到体现 (好比线性方程组的性质都会从系数行列式得到体现那样)。那么方程 “根式可解” 的话，也必定能从 “预解群” 中得到体现。

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It will be simplest to assume a strong meaning for “solvable by radicals,” ...

---- 最简单的乃是给 “根式可解” 赋予较强的意思...

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... namely, that all roots (not just one root) of the equation can be expressed in terms of known quantities...

---- 即，方程的所有根 (不只是一个根) 都能表达为已知量...

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... and the operations of addition, substraction, multiplication, division by nonzero quantities, and the extraction of roots.

---- 以及加减乘除和根的开方(提取?)之操作。

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评论：末尾的 “extraction of roots” 不易理解... 应该不是给根开方，而是通过开方提取根 (?)。

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In fact, it can be shown (Exercise 4) that if one root of an irreducible equation can be expressed in this way then all can, but this is a fine point that can wait.

---- 事实上可以看到 (Ex.4)，若不可约方程的一个根能以这种方式表达，则所有的都可以。这很好暂按不表。

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小结：开始新一节(S43)的阅读。上面是前两段 (乏味、没有张力，后面五段打算倒着读)。

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