Maxima で綴る数学の旅

紙と鉛筆の代わりに、数式処理システムMaxima / Macsyma を使って、数学を楽しみましょう

-数学- 可解な方程式を冪根で解く -可解な5次多項式の根も冪根で計算出来た-

 

元々は前回の記事:

の続きを書くつもりだったのですが、少しだけ予定を変更して、SolveSolvable()関数での5次多項式のサポート状況を書きます。例題としては\(x^5-5\,x+12\)を使います。

前回の記事を書いてから、作ったプログラムに色々と不備(後述)が見つかり、その修正を行いました。そしてそれらの修正をSolveSolvable2.macおよびその補助ファイルに反映しました。現在のバージョンでは例えば上記の5次多項式の根を冪根で求めることができます。

(%i29) load("SolveSolvable2.mac")$

では早速例題の多項式をp1という変数に設定します。

(%i30) p1:x^5-5*x+12;
$$ \tag{%o30} x^5-5\,x+12 $$

SolveSolvable()という関数で引数に変数xの多項式を与えると解を冪根で求める計算が始まります。計算途中でVの最小多項式ガロア群、組成列、組成列の隣り合う群の位数の商、全ての解、解の中に現れる冪根のリスト、数値計算による検算、が順次表示されます。
(%i31) SolveSolvable(p1)$

Vの最小多項式
$$ \tag{*} \verb|Minimal polynomial of V|\verb| | \\
 V^{10}-230\,V^8+840\,V^7-1475\,V^6-302352\,V^5+2155900\,V^4+10857480\,V^3-115816340\,V^2+2905357200\,V+23342655376 $$

ガロア
$$ \tag{*} \verb|Galois Group of|\verb| |x^5-5\,x+12\verb| | \\
\begin{pmatrix} a&b&c&d&e\\ a&c&b&e&d\\ b&a&d&c&e\\ b&d&a&e&c\\ c&a&e&b&d\\ c&e&a&d&b\\ d&b&e&a&c\\ d&e&b&c&a\\ e&c&d&a&b\\ e&d&c&b&a\\ \end{pmatrix}  $$

組成列と隣り合う群の位数の商
$$ \tag{*} \verb|Subnormal series and quotients of orders|\verb| | \\
\verb|FiniteGroup[1,8,27,38,53,68,83,94,113,120]|\verb| | \\
\verb|FiniteGroup[1,38,53,94,113]|\verb| | \\
\verb|FiniteGroup[1]|\verb| | \\
2\verb| | \\
5\verb| | $$

全ての解
$$ \tag{*} \verb|Solutions|\verb| | \\
\mathrm{\%x}_{1} =\frac{1284875787\,\alpha_{1}\,\alpha_{2}^4\,{\rm Z}_{5}^3}{34461882873770}-\frac{3694593985\,\alpha_{2}^4\,{\rm Z}_{5}^3}{202716958081}+\frac{5197513\,\alpha_{1}\,\alpha_{2}^3\,{\rm Z}_{5}^3}{51358990870}-\frac{2641711\,\alpha_{2}^3\,{\rm Z}_{5}^3}{302111711}-\frac{3182\,\alpha_{1}\,\alpha_{2}^2\,{\rm Z}_{5}^3}{191352425}+\frac{68943\,\alpha_{2}^2\,{\rm Z}_{5}^3}{450241}-\frac{42\,\alpha_{2}\,{\rm Z}_{5}^3}{671}+\frac{283621329\,\alpha_{1}\,\alpha_{2}^4\,{\rm Z}_{5}^2}{6892376574754}-\frac{406631190\,\alpha_{2}^4\,{\rm Z}_{5}^2}{202716958081}+\frac{693488\,\alpha_{1}\,\alpha_{2}^3\,{\rm Z}_{5}^2}{25679495435}-\frac{1942690\,\alpha_{2}^3\,{\rm Z}_{5}^2}{302111711}-\frac{62603\,\alpha_{1}\,\alpha_{2}^2\,{\rm Z}_{5}^2}{382704850}+\frac{7999\,\alpha_{2}^2\,{\rm Z}_{5}^2}{450241}+\frac{129\,\alpha_{2}\,{\rm Z}_{5}^2}{671}+\frac{146040951\,\alpha_{1}\,\alpha_{2}^4\,{\rm Z}_{5}}{6892376574754}-\frac{2440114390\,\alpha_{2}^4\,{\rm Z}_{5}}{202716958081}+\frac{1959806\,\alpha_{1}\,\alpha_{2}^3\,{\rm Z}_{5}}{25679495435}+\frac{3363898\,\alpha_{2}^3\,{\rm Z}_{5}}{302111711}+\frac{16109\,\alpha_{1}\,\alpha_{2}^2\,{\rm Z}_{5}}{191352425}+\frac{20703\,\alpha_{2}^2\,{\rm Z}_{5}}{450241}-\frac{40\,\alpha_{2}\,{\rm Z}_{5}}{671}+\frac{2083525533\,\alpha_{1}\,\alpha_{2}^4}{34461882873770}-\frac{114074985\,\alpha_{2}^4}{202716958081}+\frac{86551\,\alpha_{1}\,\alpha_{2}^3}{1510558555}-\frac{168782\,\alpha_{2}^3}{302111711}-\frac{17533\,\alpha_{1}\,\alpha_{2}^2}{191352425}+\frac{54580\,\alpha_{2}^2}{450241}+\frac{25\,\alpha_{2}}{671}\verb| | $$

$$ \tag{*} \mathrm{\%x}_{2}=\frac{66615429\,\alpha_{1}\,\alpha_{2}^4\,{\rm Z}_{5}^3}{17230941436885}+\frac{3287962795\,\alpha_{2}^4\,{\rm Z}_{5}^3}{202716958081}+\frac{1266318\,\alpha_{1}\,\alpha_{2}^3\,{\rm Z}_{5}^3}{25679495435}+\frac{5306588\,\alpha_{2}^3\,{\rm Z}_{5}^3}{302111711}-\frac{33642\,\alpha_{1}\,\alpha_{2}^2\,{\rm Z}_{5}^3}{191352425}+\frac{33877\,\alpha_{2}^2\,{\rm Z}_{5}^3}{450241}-\frac{25\,\alpha_{2}\,{\rm Z}_{5}^3}{671}-\frac{277335516\,\alpha_{1}\,\alpha_{2}^4\,{\rm Z}_{5}^2}{17230941436885}+\frac{1254479595\,\alpha_{2}^4\,{\rm Z}_{5}^2}{202716958081}+\frac{777879\,\alpha_{1}\,\alpha_{2}^3\,{\rm Z}_{5}^2}{25679495435}+\frac{1773908\,\alpha_{2}^3\,{\rm Z}_{5}^2}{302111711}-\frac{16109\,\alpha_{1}\,\alpha_{2}^2\,{\rm Z}_{5}^2}{191352425}-\frac{20703\,\alpha_{2}^2\,{\rm Z}_{5}^2}{450241}-\frac{67\,\alpha_{2}\,{\rm Z}_{5}^2}{671}+\frac{399324873\,\alpha_{1}\,\alpha_{2}^4\,{\rm Z}_{5}}{17230941436885}+\frac{3580519000\,\alpha_{2}^4\,{\rm Z}_{5}}{202716958081}-\frac{693488\,\alpha_{1}\,\alpha_{2}^3\,{\rm Z}_{5}}{25679495435}+\frac{1942690\,\alpha_{2}^3\,{\rm Z}_{5}}{302111711}-\frac{19291\,\alpha_{1}\,\alpha_{2}^2\,{\rm Z}_{5}}{191352425}+\frac{48240\,\alpha_{2}^2\,{\rm Z}_{5}}{450241}+\frac{104\,\alpha_{2}\,{\rm Z}_{5}}{671}-\frac{1284875787\,\alpha_{1}\,\alpha_{2}^4}{34461882873770}+\frac{3694593985\,\alpha_{2}^4}{202716958081}+\frac{3810537\,\alpha_{1}\,\alpha_{2}^3}{51358990870}-\frac{699021\,\alpha_{2}^3}{302111711}-\frac{94821\,\alpha_{1}\,\alpha_{2}^2}{382704850}-\frac{12704\,\alpha_{2}^2}{450241}-\frac{65\,\alpha_{2}}{671}\verb| | \\ $$

$$ \tag{*} \mathrm{\%x}_{3}=-\frac{2083525533\,\alpha_{1}\,\alpha_{2}^4\,{\rm Z}_{5}^3}{34461882873770}+\frac{114074985\,\alpha_{2}^4\,{\rm Z}_{5}^3}{202716958081}-\frac{488439\,\alpha_{1}\,\alpha_{2}^3\,{\rm Z}_{5}^3}{25679495435}-\frac{3532680\,\alpha_{2}^3\,{\rm Z}_{5}^3}{302111711}+\frac{94821\,\alpha_{1}\,\alpha_{2}^2\,{\rm Z}_{5}^3}{382704850}+\frac{12704\,\alpha_{2}^2\,{\rm Z}_{5}^3}{450241}+\frac{171\,\alpha_{2}\,{\rm Z}_{5}^3}{671}-\frac{399324873\,\alpha_{1}\,\alpha_{2}^4\,{\rm Z}_{5}^2}{17230941436885}-\frac{3580519000\,\alpha_{2}^4\,{\rm Z}_{5}^2}{202716958081}-\frac{1959806\,\alpha_{1}\,\alpha_{2}^3\,{\rm Z}_{5}^2}{25679495435}-\frac{3363898\,\alpha_{2}^3\,{\rm Z}_{5}^2}{302111711}+\frac{27537\,\alpha_{1}\,\alpha_{2}^2\,{\rm Z}_{5}^2}{382704850}+\frac{46581\,\alpha_{2}^2\,{\rm Z}_{5}^2}{450241}+\frac{2\,\alpha_{2}\,{\rm Z}_{5}^2}{671}-\frac{332709444\,\alpha_{1}\,\alpha_{2}^4\,{\rm Z}_{5}}{17230941436885}-\frac{292556205\,\alpha_{2}^4\,{\rm Z}_{5}}{202716958081}+\frac{1277901\,\alpha_{1}\,\alpha_{2}^3\,{\rm Z}_{5}}{51358990870}-\frac{6005609\,\alpha_{2}^3\,{\rm Z}_{5}}{302111711}+\frac{62603\,\alpha_{1}\,\alpha_{2}^2\,{\rm Z}_{5}}{382704850}-\frac{7999\,\alpha_{2}^2\,{\rm Z}_{5}}{450241}+\frac{67\,\alpha_{2}\,{\rm Z}_{5}}{671}-\frac{676660389\,\alpha_{1}\,\alpha_{2}^4}{17230941436885}-\frac{2326039405\,\alpha_{2}^4}{202716958081}-\frac{1266318\,\alpha_{1}\,\alpha_{2}^3}{25679495435}-\frac{5306588\,\alpha_{2}^3}{302111711}+\frac{56239\,\alpha_{1}\,\alpha_{2}^2}{382704850}+\frac{60944\,\alpha_{2}^2}{450241}+\frac{42\,\alpha_{2}}{671}\verb| | \\ $$
$$ \tag{*} \mathrm{\%x}_{4}=-\frac{68790189\,\alpha_{1}\,\alpha_{2}^4\,{\rm Z}_{5}^3}{3446188287377}-\frac{2033483200\,\alpha_{2}^4\,{\rm Z}_{5}^3}{202716958081}-\frac{86551\,\alpha_{1}\,\alpha_{2}^3\,{\rm Z}_{5}^3}{1510558555}+\frac{168782\,\alpha_{2}^3\,{\rm Z}_{5}^3}{302111711}-\frac{56239\,\alpha_{1}\,\alpha_{2}^2\,{\rm Z}_{5}^3}{382704850}-\frac{60944\,\alpha_{2}^2\,{\rm Z}_{5}^3}{450241}+\frac{65\,\alpha_{2}\,{\rm Z}_{5}^3}{671}+\frac{332709444\,\alpha_{1}\,\alpha_{2}^4\,{\rm Z}_{5}^2}{17230941436885}+\frac{292556205\,\alpha_{2}^4\,{\rm Z}_{5}^2}{202716958081}+\frac{2254779\,\alpha_{1}\,\alpha_{2}^3\,{\rm Z}_{5}^2}{51358990870}-\frac{2472929\,\alpha_{2}^3\,{\rm Z}_{5}^2}{302111711}+\frac{19291\,\alpha_{1}\,\alpha_{2}^2\,{\rm Z}_{5}^2}{191352425}-\frac{48240\,\alpha_{2}^2\,{\rm Z}_{5}^2}{450241}+\frac{40\,\alpha_{2}\,{\rm Z}_{5}^2}{671}-\frac{283621329\,\alpha_{1}\,\alpha_{2}^4\,{\rm Z}_{5}}{6892376574754}+\frac{406631190\,\alpha_{2}^4\,{\rm Z}_{5}}{202716958081}-\frac{777879\,\alpha_{1}\,\alpha_{2}^3\,{\rm Z}_{5}}{25679495435}-\frac{1773908\,\alpha_{2}^3\,{\rm Z}_{5}}{302111711}-\frac{14351\,\alpha_{1}\,\alpha_{2}^2\,{\rm Z}_{5}}{191352425}-\frac{14363\,\alpha_{2}^2\,{\rm Z}_{5}}{450241}-\frac{2\,\alpha_{2}\,{\rm Z}_{5}}{671}-\frac{66615429\,\alpha_{1}\,\alpha_{2}^4}{17230941436885}-\frac{3287962795\,\alpha_{2}^4}{202716958081}+\frac{488439\,\alpha_{1}\,\alpha_{2}^3}{25679495435}+\frac{3532680\,\alpha_{2}^3}{302111711}+\frac{3182\,\alpha_{1}\,\alpha_{2}^2}{191352425}-\frac{68943\,\alpha_{2}^2}{450241}+\frac{169\,\alpha_{2}}{671}\verb| | \\ $$
$$ \tag{*} \mathrm{\%x}_{5}=\frac{676660389\,\alpha_{1}\,\alpha_{2}^4\,{\rm Z}_{5}^3}{17230941436885}+\frac{2326039405\,\alpha_{2}^4\,{\rm Z}_{5}^3}{202716958081}-\frac{3810537\,\alpha_{1}\,\alpha_{2}^3\,{\rm Z}_{5}^3}{51358990870}+\frac{699021\,\alpha_{2}^3\,{\rm Z}_{5}^3}{302111711}+\frac{17533\,\alpha_{1}\,\alpha_{2}^2\,{\rm Z}_{5}^3}{191352425}-\frac{54580\,\alpha_{2}^2\,{\rm Z}_{5}^3}{450241}-\frac{169\,\alpha_{2}\,{\rm Z}_{5}^3}{671}-\frac{146040951\,\alpha_{1}\,\alpha_{2}^4\,{\rm Z}_{5}^2}{6892376574754}+\frac{2440114390\,\alpha_{2}^4\,{\rm Z}_{5}^2}{202716958081}-\frac{1277901\,\alpha_{1}\,\alpha_{2}^3\,{\rm Z}_{5}^2}{51358990870}+\frac{6005609\,\alpha_{2}^3\,{\rm Z}_{5}^2}{302111711}+\frac{14351\,\alpha_{1}\,\alpha_{2}^2\,{\rm Z}_{5}^2}{191352425}+\frac{14363\,\alpha_{2}^2\,{\rm Z}_{5}^2}{450241}-\frac{104\,\alpha_{2}\,{\rm Z}_{5}^2}{671}+\frac{277335516\,\alpha_{1}\,\alpha_{2}^4\,{\rm Z}_{5}}{17230941436885}-\frac{1254479595\,\alpha_{2}^4\,{\rm Z}_{5}}{202716958081}-\frac{2254779\,\alpha_{1}\,\alpha_{2}^3\,{\rm Z}_{5}}{51358990870}+\frac{2472929\,\alpha_{2}^3\,{\rm Z}_{5}}{302111711}-\frac{27537\,\alpha_{1}\,\alpha_{2}^2\,{\rm Z}_{5}}{382704850}-\frac{46581\,\alpha_{2}^2\,{\rm Z}_{5}}{450241}-\frac{129\,\alpha_{2}\,{\rm Z}_{5}}{671}+\frac{68790189\,\alpha_{1}\,\alpha_{2}^4}{3446188287377}+\frac{2033483200\,\alpha_{2}^4}{202716958081}-\frac{5197513\,\alpha_{1}\,\alpha_{2}^3}{51358990870}+\frac{2641711\,\alpha_{2}^3}{302111711}+\frac{33642\,\alpha_{1}\,\alpha_{2}^2}{191352425}-\frac{33877\,\alpha_{2}^2}{450241}-\frac{171\,\alpha_{2}}{671}\verb| | $$

解の中に現れる冪根のリスト
$$ \tag{*} \verb|with|\verb| | \\
\left[ \left[ \alpha_{2} , -\frac{103407\,\alpha_{1}\,{\rm Z}_{5}^3}{2125}-\frac{36897\,{\rm Z}_{5}^3}{5}-\frac{162816\,\alpha_{1}\,{\rm Z}_{5}^2}{2125}+\frac{76253\,{\rm Z}_{5}^2}{5}-\frac{96498\,\alpha_{1}\,{\rm Z}_{5}}{2125}+36580\,{\rm Z}_{5}+\alpha_{2}^5+\frac{8127\,\alpha_{1}}{4250}+\frac{135979}{5} \right] , \left[ {\rm Z}_{5} , {\rm Z}_{5}^4+{\rm Z}_{5}^3+{\rm Z}_{5}^2+{\rm Z}_{5}+1 \right] , \left[ \alpha_{1} , \alpha_{1}^2+289000 \right] \right] \verb| | $$

数値計算による検算
$$ \tag{*} \verb|Verification|\verb| | \\
\verb|Numeric calcuration of the above solutions|\verb| | \\
\left[ -4.725954673740532 \times 10^{-14}\,i-1.842085966190126 , 1.709561043370229\,i-0.3518542408273611 , -1.709561043370314\,i-0.351854240827393 , 1.272897223922556-0.7197986814837805\,i , 0.7197986814838231\,i+1.27289722392247 \right] \verb| | \\ $$

$$ \tag{*} \verb|Numeric solutions with allroots(|\verb| |x^5-5\,x+12\verb| |\verb|)|\verb| | \\
\left[ x=0.7197986814838616\,i+1.272897223922499 , x=1.272897223922499-0.7197986814838616\,i , x=1.709561043370329\,i-0.351854240827372 , x=-1.709561043370329\,i-0.351854240827372 , x=-1.842085966190254 \right] \verb| | $$

(%i32) time(%o31);
$$ \tag{%o32} \left[ 7973.01 \right] $$

計算時間は2時間15分くらいで終わりました。メモリの最大使用量は計測できませんでした。

 

主な不具合は、ExtendedField.macの中で多項式同士の商を分数式ではなく多項式で求める関数、Stages.macの中でbという変数の値を求める部分にあり、

文献3 「退職後は素人数学者」可解な代数方程式のガロア理論に基づいた解法

の中の該当プログラムを読み込むことで問題を解決することができました。改めて著者の方に感謝いたします。

This is a test.
天気