[過去ﾛｸﾞ] ガロア第一論文及びその関連の資料スレ (1002ﾚｽ)
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107(2): 2023/01/31(火)11:59 ID:tkHk7/Du(1/5) AAS
>>106
１）他人が、何をどこまで理解しているか？
それを計る客観的基準はない
例えば、数学科修士の入学試験があったとして
それは、あくまで一つの指標でしかない
２）かつ、院試で満点で合格する人はおそらくいない！
合格点が何点か知らない
省7
109(1): 2023/01/31(火)12:39 ID:tkHk7/Du(2/5) AAS
>>108
>>107より
（引用開始）
他人が、何をどこまで理解しているか？
それを計る客観的基準はない
例えば、数学科修士の入学試験があったとして
それは、あくまで一つの指標でしかない
省18
110(3): 2023/01/31(火)15:55 ID:tkHk7/Du(3/5) AAS
>>75
>ガロアより以前に置換群論において正規部分群という概念を思いついた
>という人は居ないのだろうか？
ガロアは、Chevallierへの手紙（下記）で
・正規部分群について明記している
（This is called proper decomposition：G = H + H S + H S' + ・・とG = H +TH +T'H +・・とが一致するとき）
・”If each of these groups has a prime number of permutations then the equation will be solvable by radicals; otherwise, not.”と明記している
省13
111(1): 2023/01/31(火)15:56 ID:tkHk7/Du(4/5) AAS
>>110
つづき
In both the cases, the group of the equation can be partitioned by adjunction into groups such that one can pass from one to another by a self-transformation;but the condition that these groups have the same substitutions holds only in thesecond case.
This is called proper decomposition.
In other words, when a group G contains another, H, the group G can be parti-tioned into groups each of which is obtained by operating on the permutations inH a self-transformation, in such a way that,G = H + H S + H S' + ・・..And we can also partition into groups which have all similar substitutions, such that G = H +TH +T'H +・・
These two types of decompositions generally do not coincide. When they do coin-cide the decomposition is said to be proper.
It is easy to see that, when the group of an equation is not susceptible to any proper decomposition, however well we might have transformed this equation, the groupsof the transformed equations will always have the same number of permutations.On the contrary, when the group of an equation is susceptible to a proper decom-position in such a way that we can decompose it into M groups of N permutations,we can resolve the given equation by means of two equations: one will have a groupof M permutations and the other, one of N permutations.
省1
112: 2023/01/31(火)15:57 ID:tkHk7/Du(5/5) AAS
>>111
つづき
Hence when we would have exhausted all possible proper decompositions on the group of an equation, we arrive at groups which can be transformed but for whichthe number of permutations will always be the same.
If each of these groups has a prime number of permutations then the equation will be solvable by radicals; otherwise, not.
The smallest number of permutations that an indecomposable group can have,when this number is not a prime number, is 5・4・3.（＝60のA5(交代群)）
(引用終り)
以上
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