>>153 補足追加

１．ｎ次方程式、ｎ個の根を持つ。代数学の基本定理から、ｎ個の根は複素数の範囲だ。だが、周知のように、有理数の範囲ではない（無理数）
２．ｎ個の根の置換の成す群が、方程式の根のもつ性質を表す。これぞ、ガロア理論の思想でしょ？
３．べき根拡大 英 Radical extension と 巡回拡大（つまり ガロア群が巡回群のとき）下記
　そして、以下のような
　”A field extension is called a cyclic extension if its Galois group is cyclic.
　For fields of characteristic zero, such extensions are the subject of Kummer theory, and are intimately related to solvability by radicals.”
　ここをちゃんと語らないと（＾＾
４．ガロア理論を語ったことにならない！（海城 >>139を見よ！！（＾＾ ）

（参考）
https://en.wikipedia.org/wiki/Radical_extension
Radical extension

Solvability by radicals
The proof is related to Lagrange resolvents.

It follows from this theorem that a Galois extension may be expressed as a radical series if and only if its Galois group is solvable.
This is, in modern terminology, the criterion of solvability by radicals that was provided by Galois.
The proof uses the fact that the Galois closure of a simple radical extension of degree n is the extension of it by a primitive nth root of unity, and that the Galois group of the nth roots of unity is cyclic.

https://ja.wikipedia.org/wiki/%E3%82%A2%E3%83%BC%E3%83%99%E3%83%AB%E6%8B%A1%E5%A4%A7
アーベル拡大

つづく