数学\(\mathtt{ VB } \ \)ガロア流方程式の解法技術
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【3-5】 写像 \(\rho_i\) が群構造を持つことの確認
\(F_0(v)\) は \(F_0\) のガロア拡大である事がわかったので、ガロア理論では次の定理が成り立ちます。「 \(F_0(v)\) には原始元 \(v\) を \(g_0(x)\) の共役根に変換する自己同型写像が共役根の数だけ存在して、 この自己同型写像は群をなす。」
この節ではこの定理が成り立っていることを確かめます。
その為に、(5.1)で定義される \( \{\rho_i\} \ [i=1,2,..,24]\) を自己同型写像の候補として考える事にします。
この(5.1)は、(4.10-13)の \(v_i\) の多項式表現を写像 \(\rho_i\) だとみなしたものです。
以下この \(\rho_i\) が対称群 \(S_4\) 群をなし、かつ共役根同士の自己同型写像になっている事を確認します。
\begin{align} \rho_{1}(v) &\equiv v_1=v \notag \\ \notag \\ \rho_{2}(v) &\equiv v_2=\frac{801167701943012874015343807}{2531636596706154203109159208286706204672}v^{23}-\frac{51207699710669004924125}{99737485589022345786910893443907584}v^{22} \notag \\ &\quad .......... \notag \\ &+\frac{4663968749846627680850109200993107}{2971516062464968406218495540075104}v+\frac{2718803338720300088760700554765}{1521873254227025539961408896544} \\ \notag \\ &\qquad \qquad .........\notag \\ \notag \\ \rho_{23}(v) &\equiv v_{23}=-\frac{801167701943012874015343807}{2531636596706154203109159208286706204672}v^{23}-\frac{51207699710669004924125}{99737485589022345786910893443907584}v^{22} \notag \\ &\quad .......... \notag \\ &-\frac{4663968749846627680850109200993107}{2971516062464968406218495540075104}v+\frac{2718803338720300088760700554765}{1521873254227025539961408896544} \notag \\ \notag \\ \rho_{24}(v) &\equiv v_{24}=-v \notag \\ \end{align}
(5.1)で定義された \(\rho_i(v), \ [i=1,2,...,24]\) が同型写像として群をなすことを確認する必要があります。 その為には、 \(\rho_i \circ \rho_j = \rho_k\) の積表を作成して、同型写像同士の積が群をなしているかどうかを検証しなければなりません。 検証の手続きは【第2章】の【2-6】と全く同様にします。第3章では \(v_i\) の式が長くなりすぎて計算過程を記述できませんが、 計算の結果の積表を下に示します。
これより写像 \(\rho_i\)は積に対して、対称群 \(S_4\) と等価な群を構成している事が判ります
| \( i \backslash j \) | \(1 \) | \(2 \) | \(3 \) | \(4 \) | \(5 \) | \(6 \) | \(7 \) | \(8 \) | \(9 \) | \({10} \) | \({11} \) | \({12} \) | \({13} \) | \({14} \) | \({15} \) | \({16} \) | \({17} \) | \({18} \) | \({19} \) | \({20} \) | \({21} \) | \({22} \) | \({23} \) | \({24}\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(1\) | \(1 \) | \(2 \) | \(3 \) | \(4 \) | \(5 \) | \(6 \) | \(7 \) | \(8 \) | \(9 \) | \({10} \) | \({11} \) | \({12} \) | \({13} \) | \({14} \) | \({15} \) | \({16} \) | \({17} \) | \({18} \) | \({19} \) | \({20} \) | \({21} \) | \({22} \) | \({23} \) | \({24}\) |
| \(2\) | \(2 \) | \(1 \) | \(5 \) | \(6 \) | \(3 \) | \(4 \) | \(8 \) | \(7 \) | \({11} \) | \({12} \) | \(9 \) | \({10} \) | \({19} \) | \({20} \) | \({21} \) | \({22} \) | \({23} \) | \({24}\) | \({13} \) | \({14} \) | \({15} \) | \({16} \) | \({17} \) | \({18}\) |
| \(3 \) | \(3 \) | \(4 \) | \(1 \) | \(2 \) | \(6 \) | \(5 \) | \({13} \) | \({14} \) | \({15} \) | \({16} \) | \({17} \) | \({18} \) | \(7 \) | \(8 \) | \(9 \) | \({10} \) | \({11} \) | \({12} \) | \({20} \) | \({19} \) | \({23} \) | \({24}\) | \({21} \) | \({22} \) |
| \(4\) | \(4 \) | \(3 \) | \(6 \) | \(5 \) | \(1 \) | \(2 \) | \({14} \) | \({13} \) | \({17} \) | \({18} \) | \({15} \) | \({16} \) | \({20} \) | \({19} \) | \({23} \) | \({24}\) | \({21} \) | \({22} \) | \(7 \) | \(8 \) | \(9 \) | \({10} \) | \({11} \) | \({12} \) |
| \(5 \) | \(5 \) | \(6 \) | \(2 \) | \(1 \) | \(4 \) | \(3 \) | \({19} \) | \({20} \) | \({21} \) | \({22} \) | \({23} \) | \({24}\) | \(8 \) | \(7 \) | \({11} \) | \({12} \) | \(9 \) | \({10} \) | \({14} \) | \({13} \) | \({17} \) | \({18} \) | \({15} \) | \({16} \) |
| \(6 \) | \(6 \) | \(5 \) | \(4 \) | \(3 \) | \(2 \) | \(1 \) | \({20} \) | \({19} \) | \({23} \) | \({24}\) | \({21} \) | \({22} \) | \({14} \) | \({13} \) | \({17} \) | \({18} \) | \({15} \) | \({16} \) | \(8 \) | \(7 \) | \({11} \) | \({12} \) | \(9 \) | \({10} \) |
| \(7 \) | \(7 \) | \(8 \) | \(9 \) | \({10} \) | \({11} \) | \({12} \) | \(1 \) | \(2 \) | \(3 \) | \(4 \) | \(5 \) | \(6 \) | \({15} \) | \({16} \) | \({13} \) | \({14} \) | \({18} \) | \({17} \) | \({21} \) | \({22} \) | \({19} \) | \({20} \) | \({24}\) | \({23} \) |
| \(8 \) | \(8 \) | \(7 \) | \({11} \) | \({12} \) | \(9 \) | \({10} \) | \(2 \) | \(1 \) | \(5 \) | \(6 \) | \(3 \) | \(4 \) | \({21} \) | \({22} \) | \({19} \) | \({20} \) | \({24}\) | \({23} \) | \({15} \) | \({16} \) | \({13} \) | \({14} \) | \({18} \) | \({17} \) |
| \(9 \) | \(9 \) | \({10} \) | \(7 \) | \(8 \) | \({12} \) | \({11} \) | \({15} \) | \({16} \) | \({13} \) | \({14} \) | \({18} \) | \({17} \) | \(1 \) | \(2 \) | \(3 \) | \(4 \) | \(5 \) | \(6 \) | \({22} \) | \({21} \) | \({24}\) | \({23} \) | \({19} \) | \({20} \) |
| \({10} \) | \({10} \) | \(9 \) | \({12} \) | \({11} \) | \(7 \) | \(8 \) | \({16} \) | \({15} \) | \({18} \) | \({17} \) | \({13} \) | \({14} \) | \({22} \) | \({21} \) | \({24}\) | \({23} \) | \({19} \) | \({20} \) | \(1 \) | \(2 \) | \(3 \) | \(4 \) | \(5 \) | \(6 \) |
| \({11} \) | \({11} \) | \({12} \) | \(8 \) | \(7 \) | \({10} \) | \(9 \) | \({21} \) | \({22} \) | \({19} \) | \({20} \) | \({24}\) | \({23} \) | \(2 \) | \(1 \) | \(5 \) | \(6 \) | \(3 \) | \(4 \) | \({16} \) | \({15} \) | \({18} \) | \({17} \) | \({13} \) | \({14} \) |
| \({12} \) | \({12} \) | \({11} \) | \({10} \) | \(9 \) | \(8 \) | \(7 \) | \({22} \) | \({21} \) | \({24}\) | \({23} \) | \({19} \) | \({20} \) | \({16} \) | \({15} \) | \({18} \) | \({17} \) | \({13} \) | \({14} \) | \(2 \) | \(1 \) | \(5 \) | \(6 \) | \(3 \) | \(4 \) |
| \({13} \) | \({13} \) | \({14} \) | \({15} \) | \({16} \) | \({17} \) | \({18} \) | \(3 \) | \(4 \) | \(1 \) | \(2 \) | \(6 \) | \(5 \) | \(9 \) | \({10} \) | \(7 \) | \(8 \) | \({12} \) | \({11} \) | \({23} \) | \({24}\) | \({20} \) | \({19} \) | \({22} \) | \({21} \) |
| \({14} \) | \({14} \) | \({13} \) | \({17} \) | \({18} \) | \({15} \) | \({16} \) | \(4 \) | \(3 \) | \(6 \) | \(5 \) | \(1 \) | \(2 \) | \({23} \) | \({24}\) | \({20} \) | \({19} \) | \({22} \) | \({21} \) | \(9 \) | \({10} \) | \(7 \) | \(8 \) | \({12} \) | \({11} \) |
| \({15} \) | \({15} \) | \({16} \) | \({13} \) | \({14} \) | \({18} \) | \({17} \) | \(9 \) | \({10} \) | \(7 \) | \(8 \) | \({12} \) | \({11} \) | \(3 \) | \(4 \) | \(1 \) | \(2 \) | \(6 \) | \(5 \) | \({24}\) | \({23} \) | \({22} \) | \({21} \) | \({20} \) | \({19} \) |
| \({16} \) | \({16} \) | \({15} \) | \({18} \) | \({17} \) | \({13} \) | \({14} \) | \({10} \) | \(9 \) | \({12} \) | \({11} \) | \(7 \) | \(8 \) | \({24}\) | \({23} \) | \({22} \) | \({21} \) | \({20} \) | \({19} \) | \(3 \) | \(4 \) | \(1 \) | \(2 \) | \(6 \) | \(5 \) |
| \({17} \) | \({17} \) | \({18} \) | \({14} \) | \({13} \) | \({16} \) | \({15} \) | \({23} \) | \({24}\) | \({20} \) | \({19} \) | \({22} \) | \({21} \) | \(4 \) | \(3 \) | \(6 \) | \(5 \) | \(1 \) | \(2 \) | \({10} \) | \(9 \) | \({12} \) | \({11} \) | \(7 \) | \(8 \) |
| \({18} \) | \({18} \) | \({17} \) | \({16} \) | \({15} \) | \({14} \) | \({13} \) | \({24}\) | \({23} \) | \({22} \) | \({21} \) | \({20} \) | \({19} \) | \({10} \) | \(9 \) | \({12} \) | \({11} \) | \(7 \) | \(8 \) | \(4 \) | \(3 \) | \(6 \) | \(5 \) | \(1 \) | \(2 \) |
| \({19} \) | \({19} \) | \({20} \) | \({21} \) | \({22} \) | \({23} \) | \({24}\) | \(5 \) | \(6 \) | \(2 \) | \(1 \) | \(4 \) | \(3 \) | \({11} \) | \({12} \) | \(8 \) | \(7 \) | \({10} \) | \(9 \) | \({17} \) | \({18} \) | \({14} \) | \({13} \) | \({16} \) | \({15} \) |
| \({20} \) | \({20} \) | \({19} \) | \({23} \) | \({24}\) | \({21} \) | \({22} \) | \(6 \) | \(5 \) | \(4 \) | \(3 \) | \(2 \) | \(1 \) | \({17} \) | \({18} \) | \({14} \) | \({13} \) | \({16} \) | \({15} \) | \({11} \) | \({12} \) | \(8 \) | \(7 \) | \({10} \) | \(9 \) |
| \({21} \) | \({21} \) | \({22} \) | \({19} \) | \({20} \) | \({24}\) | \({23} \) | \({11} \) | \({12} \) | \(8 \) | \(7 \) | \({10} \) | \(9 \) | \(5 \) | \(6 \) | \(2 \) | \(1 \) | \(4 \) | \(3 \) | \({18} \) | \({17} \) | \({16} \) | \({15} \) | \({14} \) | \({13} \) |
| \({22} \) | \({22} \) | \({21} \) | \({24}\) | \({23} \) | \({19} \) | \({20} \) | \({12} \) | \({11} \) | \({10} \) | \(9 \) | \(8 \) | \(7 \) | \({18} \) | \({17} \) | \({16} \) | \({15} \) | \({14} \) | \({13} \) | \(5 \) | \(6 \) | \(2 \) | \(1 \) | \(4 \) | \(3 \) |
| \({23} \) | \({23} \) | \({24}\) | \({20} \) | \({19} \) | \({22} \) | \({21} \) | \({17} \) | \({18} \) | \({14} \) | \({13} \) | \({16} \) | \({15} \) | \(6 \) | \(5 \) | \(4 \) | \(3 \) | \(2 \) | \(1 \) | \({12} \) | \({11} \) | \({10} \) | \(9 \) | \(8 \) | \(7 \) |
| \({24}\) | \({24}\) | \({23} \) | \({22} \) | \({21} \) | \({20} \) | \({19} \) | \({18} \) | \({17} \) | \({16} \) | \({15} \) | \({14} \) | \({13} \) | \({12} \) | \({11} \) | \({10} \) | \(9 \) | \(8 \) | \(7 \) | \(6 \) | \(5 \) | \(4 \) | \(3 \) | \(2 \) | \(1 \) |
【第2章】では、上記以外に \(\rho_i(v_j)\) の表も記述しましたが、表の中身は全く同一で、上記の表の数値を読み替えてもらえば 十分です。 結論として、【表3-1】は、対称群 \(S_4\) と等価な群を構成している事が判ります。さらに写像 \(\rho_i\) は、 全ての \(v_j\) に対して、 \(\rho_i(v_j)=v_k \) という写像関係を有している事も判ります。
従って自己同型写像候補であった \(\{\rho_i\}\) は、ガロア群 \(Gal(F_0(v)/F_0) \) の元そのものである事が判りました。
【3-6】写像 \(\rho_i\) の \(\{\alpha,\beta,\gamma,\delta\}\) に対する置換作用
\( \{ \rho_1,\rho_2,..,\rho_{24}\}\) が、4根に対してどの様に作用するのか計算してみます。この計算も 式が複雑になっていしまうので省略します。但し計算方法は第2章と全く同様です。全ての \(\rho_i\) が、 \([\ \alpha,\beta,\gamma,\delta \ ]\) に対してどの様に作用するか 計算してみたのが【表3-2】です。
写像 \(\rho_i\) は(2.2)の \(S_4\) の元 \(\sigma_i\) が \([\ \alpha,\beta,\gamma,\delta \ ]\) に対する置換操作と 全く同一である事が判ります。
| \(\rho_i(\alpha)\) | \(\rho_i(\beta)\) | \(\rho_i(\gamma)\) | \(\rho_i(\delta)\) | |
|---|---|---|---|---|
| \(\rho_{1}\) | \(\alpha\) | \(\beta\) | \(\gamma\) | \(\delta\) |
| \(\rho_{2}\) | \(\alpha\) | \(\beta\) | \(\delta\) | \(\gamma\) |
| \(\rho_{3}\) | \(\alpha\) | \(\gamma\) | \(\beta\) | \(\delta\) |
| \(\rho_{4}\) | \(\alpha\) | \(\gamma\) | \(\delta\) | \(\beta\) |
| \(\rho_{5}\) | \(\alpha\) | \(\delta\) | \(\beta\) | \(\gamma\) |
| \(\rho_{6}\) | \(\alpha\) | \(\delta\) | \(\gamma\) | \(\beta\) |
| \(\rho_{7}\) | \(\beta\) | \(\alpha\) | \(\gamma\) | \(\delta\) |
| \(\rho_{8}\) | \(\beta\) | \(\alpha\) | \(\delta\) | \(\gamma\) |
| \(\rho_i(\alpha)\) | \(\rho_i(\beta)\) | \(\rho_i(\gamma)\) | \(\rho_i(\delta)\) | |
|---|---|---|---|---|
| \(\rho_{9}\) | \(\beta\) | \(\gamma\) | \(\alpha\) | \(\delta\) |
| \(\rho_{10}\) | \(\beta\) | \(\gamma\) | \(\delta\) | \(\alpha\) |
| \(\rho_{11}\) | \(\beta\) | \(\delta\) | \(\alpha\) | \(\gamma\) |
| \(\rho_{12}\) | \(\beta\) | \(\delta\) | \(\gamma\) | \(\alpha\) |
| \(\rho_{13}\) | \(\gamma\) | \(\alpha\) | \(\beta\) | \(\delta\) |
| \(\rho_{14}\) | \(\gamma\) | \(\alpha\) | \(\delta\) | \(\beta\) |
| \(\rho_{15}\) | \(\gamma\) | \(\beta\) | \(\alpha\) | \(\delta\) |
| \(\rho_{16}\) | \(\gamma\) | \(\beta\) | \(\delta\) | \(\alpha\) |
| \(\rho_i(\alpha)\) | \(\rho_i(\beta)\) | \(\rho_i(\gamma)\) | \(\rho_i(\delta)\) | \(\rho_{17}\) | \(\gamma\) | \(\delta\) | \(\alpha\) | \(\beta\) |
|---|---|---|---|---|
| \(\rho_{18}\) | \(\gamma\) | \(\delta\) | \(\beta\) | \(\alpha\) | \(\rho_{19}\) | \(\delta\) | \(\alpha\) | \(\beta\) | \(\gamma\) |
| \(\rho_{20}\) | \(\delta\) | \(\alpha\) | \(\gamma\) | \(\beta\) |
| \(\rho_{21}\) | \(\delta\) | \(\beta\) | \(\alpha\) | \(\gamma\) |
| \(\rho_{22}\) | \(\delta\) | \(\beta\) | \(\gamma\) | \(\alpha\) |
| \(\rho_{23}\) | \(\delta\) | \(\gamma\) | \(\alpha\) | \(\beta\) |
| \(\rho_{24}\) | \(\delta\) | \(\gamma\) | \(\beta\) | \(\alpha\) |
本来写像 \(\rho_i\) は \(v\) を共役根に写像するものであったのですが、その写像機能は元の方程式の根 \(\{\ \alpha,\beta,\gamma,\delta \ \}\)に対しても、写像の機能が \(v\) を通して間接的に及んでいるというところが面白いです。
とにかく写像 \(\rho_i\) は写像っぽいところがいかにも自己同型写像という名前にふさわしい気がします。 \(\sigma_i\) の置換演算子の雰囲気とは違いますね。（本当は全く同じなんですけど、、、） まとめる以下となります。
\( \quad\{\rho_1,\rho_2,..,\rho_{24}\} = Gal(F_0(v)/F_0) \cong S_4\)