円分体上の Galois 群 - ATPとCASのこと

前回までの冪根拡大では，円分体を基礎体とする一方，中間体の生成は $\mathbb{Q}$ 上の Galois 群の組成列に従っており， $M_i(X,0)\in F_{i-1}[X]$ となる $i$ の発生原因となっていました．今回はこれを「まず円分体とその上の分解体の Galois 群を構成し，その組成列を用いて中間体を生成する手順」に，また，「最小多項式の文字列としてのサイズに応じて， $M_i(X,0)$ の展開方法を切り替える方式」（下の表示に出力があるものは各外部ツール，その他は組み込み関数を使用）に改めましたので，サンプルの実行結果を公開します．

表示は
・サンプル番号
・入力多項式
・ $\mathbb{Q}$ 上の分解体の primitive element の $\mathbb{Q}$ 上の最小多項式の次数
・同じく円分体上の最小多項式の（主係数についての）次数
・各中間体の拡大次数のリスト
・組成列までの処理時間（秒）
・冪根拡大の処理時間（秒）
のリストです．

展開の外部ツールとして Nemo を用いた場合．

[1,x^2-2,2,2,[2],0.152,0.002] 
[2,x^3-3*x-1,3,3,[3],0.137,0.003] 
[3,x^4-2,8,8,[2,2,2],0.184,0.006] 
[4,x^4+x^2-1,8,8,[2,2,2],0.163,0.007] 
[5,x^4-2*x^3+2*x^2+2,12,12,[3,2,2],0.181,0.03] 
[6,x^4+2*x^3+3*x^2+4*x+5,24,24,[2,3,2,2],0.275,0.125] 
[7,x^4+x+1,24,24,[2,3,2,2],0.234,0.089] 
[8,x^5-2,20,5,[5],0.209,0.002] 
[9,x^5-5*x+12,10,10,[2,5],0.205,0.022] 
[10,x^5+20*x+32,10,10,[2,5],0.192,0.027] 
[11,x^5+11*x+44,10,10,[2,5],0.166,0.026] 
[12,x^5+x^4-4*x^3-3*x^2+3*x+1,5,5,[5],0.172,0.009] 
[13,x^5+100*x^2+1000,20,5,[5],0.21,0.012] 
[14,x^6+x^3+1,6,3,[3],0.144,0.003] 
[15,x^6-2,12,6,[2,3],0.184,0.004] 
[16,x^5-5*x^3+5*x-5,20,10,[2,5],0.19,0.022] 
[17,x^6+x^3+7,18,9,[3,3],0.166,0.011] 
[18,x^6-3*x^4+1,24,24,[2,3,2,2],0.266,0.061] 
[19,x^6+x^4-9,24,24,[2,3,2,2],0.219,0.083] 
[20,x^6+x^4-8,48,48,[2,2,3,2,2],0.611,0.703] 
[21,x^7+7*x^3+7*x^2+7*x-1,14,7,[7],0.184,0.039] 
[22,x^7-14*x^5+56*x^3-56*x+22,21,7,[7],0.19,0.033] 
[23,x^7-2,42,7,[7],0.21,0.004] 
[24,x^8-2*x^6-x^4+7*x^2-5*x+1,16,16,[2,2,2,2],0.28,0.034] 
[25,x^9+x^8+3*x^6+3*x^3-3*x^2+5*x-1,18,18,[2,3,3],0.253,0.049] 
[26,x^12-3,24,12,[2,2,3],0.516,0.009] 
[27,x^7-14*x^5+56*x^3-56*x+22,21,7,[7],0.192,0.029] 
[28,
 x^15-x^14-14*x^13+13*x^12+78*x^11-66*x^10-220*x^9+165*x^8+330*x^7-210*x^6
     -252*x^5+126*x^4+84*x^3-28*x^2-8*x+1,15,15,[3,5],0.606,0.182]
  
<permutation group with 72 generators>
Welcome to Nemo version 0.13.5
  0.634780 seconds (798.89 k allocations: 42.184 MiB, 1.88% gc time)
  3.561398 seconds (3.36 M allocations: 1014.657 MiB, 2.55% gc time)
[29,x^6+x^4-x^2+5*x-5,72,72,[2,2,2,3,3],4.496,15.317] 
  *** nfisincl: Warning: increasing stack size to 16000000.
  *** write: Warning: increasing stack size to 16000000.
<permutation group with 13 generators>
Welcome to Nemo version 0.13.5
  0.729594 seconds (834.13 k allocations: 43.711 MiB, 8.42% gc time)
  0.795107 seconds (1.01 M allocations: 50.953 MiB, 1.18% gc time)
[30,2*(x+1)^13-x^13,156,13,[13],12.036,12.616] 
  *** nfisincl: Warning: increasing stack size to 16000000.
<permutation group with 49 generators>
Welcome to Nemo version 0.13.5
  0.632685 seconds (811.09 k allocations: 42.324 MiB, 9.44% gc time)
  0.994494 seconds (1.16 M allocations: 75.809 MiB, 1.63% gc time)
Welcome to Nemo version 0.13.5
  0.639144 seconds (811.09 k allocations: 42.326 MiB, 9.76% gc time)
  0.808816 seconds (1.01 M allocations: 51.011 MiB, 1.18% gc time)
[31,
 x^14+28*x^11+28*x^10-28*x^9+140*x^8+360*x^7+147*x^6+196*x^5+336*x^4-546*x^3
     -532*x^2+896*x+823,98,49,[7,7],5.772,33.549]
  
  *** nfisincl: Warning: increasing stack size to 16000000.
  *** write: Warning: increasing stack size to 16000000.
<permutation group with 75 generators>
Welcome to Nemo version 0.13.5
  0.635679 seconds (802.47 k allocations: 41.997 MiB, 1.84% gc time)
  3.301811 seconds (2.28 M allocations: 812.627 MiB, 2.59% gc time)
Welcome to Nemo version 0.13.5
  0.688764 seconds (802.47 k allocations: 41.997 MiB, 8.95% gc time)
  1.009562 seconds (1.17 M allocations: 82.609 MiB, 1.53% gc time)
Welcome to Nemo version 0.13.5
  0.690487 seconds (802.47 k allocations: 41.997 MiB, 8.90% gc time)
  0.809244 seconds (1.01 M allocations: 50.949 MiB, 1.17% gc time)
[32,
 x^15-470*x^13-305*x^12+71840*x^11+85357*x^10-4292700*x^9-3714805*x^8
     +119761820*x^7+25284495*x^6-1542190154*x^5+717324725*x^4+7178878600*x^3
     -5452953875*x^2-7998223215*x+4461221029,75,75,[3,5,5],15.793,132.7]
  
Evaluation took 180.3810 seconds (240.6020 elapsed) using 74460.010 MB.

展開の外部ツールとして Giac を用いた場合．

[1,x^2-2,2,2,[2],0.133,0.001] 
[2,x^3-3*x-1,3,3,[3],0.169,0.003] 
[3,x^4-2,8,8,[2,2,2],0.168,0.005] 
[4,x^4+x^2-1,8,8,[2,2,2],0.16,0.007] 
[5,x^4-2*x^3+2*x^2+2,12,12,[3,2,2],0.174,0.029] 
[6,x^4+2*x^3+3*x^2+4*x+5,24,24,[2,3,2,2],0.273,0.126] 
[7,x^4+x+1,24,24,[2,3,2,2],0.224,0.087] 
[8,x^5-2,20,5,[5],0.182,0.002] 
[9,x^5-5*x+12,10,10,[2,5],0.18,0.023] 
[10,x^5+20*x+32,10,10,[2,5],0.163,0.032] 
[11,x^5+11*x+44,10,10,[2,5],0.146,0.025] 
[12,x^5+x^4-4*x^3-3*x^2+3*x+1,5,5,[5],0.141,0.008] 
[13,x^5+100*x^2+1000,20,5,[5],0.161,0.012] 
[14,x^6+x^3+1,6,3,[3],0.139,0.002] 
[15,x^6-2,12,6,[2,3],0.167,0.004] 
[16,x^5-5*x^3+5*x-5,20,10,[2,5],0.171,0.023] 
[17,x^6+x^3+7,18,9,[3,3],0.162,0.009] 
[18,x^6-3*x^4+1,24,24,[2,3,2,2],0.243,0.061] 
[19,x^6+x^4-9,24,24,[2,3,2,2],0.218,0.083] 
[20,x^6+x^4-8,48,48,[2,2,3,2,2],0.612,0.702] 
[21,x^7+7*x^3+7*x^2+7*x-1,14,7,[7],0.17,0.04] 
[22,x^7-14*x^5+56*x^3-56*x+22,21,7,[7],0.175,0.028] 
[23,x^7-2,42,7,[7],0.185,0.004] 
[24,x^8-2*x^6-x^4+7*x^2-5*x+1,16,16,[2,2,2,2],0.274,0.033] 
[25,x^9+x^8+3*x^6+3*x^3-3*x^2+5*x-1,18,18,[2,3,3],0.241,0.048] 
[26,x^12-3,24,12,[2,2,3],0.52,0.009] 
[27,x^7-14*x^5+56*x^3-56*x+22,21,7,[7],0.177,0.029] 
[28,
 x^15-x^14-14*x^13+13*x^12+78*x^11-66*x^10-220*x^9+165*x^8+330*x^7-210*x^6
     -252*x^5+126*x^4+84*x^3-28*x^2-8*x+1,15,15,[3,5],0.605,0.189]
  
<permutation group with 72 generators>
// Using locale /usr/share/locale/
// ja_JP.UTF-8
// /usr/share/locale/
// giac
// UTF-8
// Maximum number of parallel threads 2
Added 0 synonyms

Evaluation time: 8.8
"Done","Done","Done","Done"
// Time 8.8
// Total time 8.8
[29,x^6+x^4-x^2+5*x-5,72,72,[2,2,2,3,3],4.448,15.771] 
  *** nfisincl: Warning: increasing stack size to 16000000.
  *** write: Warning: increasing stack size to 16000000.
<permutation group with 13 generators>
// Using locale /usr/share/locale/
// ja_JP.UTF-8
// /usr/share/locale/
// giac
// UTF-8
// Maximum number of parallel threads 2
Added 0 synonyms
"Done","Done","Done","Done"
// Time 0
// Total time 0
[30,2*(x+1)^13-x^13,156,13,[13],12.067,8.419] 
  *** nfisincl: Warning: increasing stack size to 16000000.
<permutation group with 49 generators>
// Using locale /usr/share/locale/
// ja_JP.UTF-8
// /usr/share/locale/
// giac
// UTF-8
// Maximum number of parallel threads 2
Added 0 synonyms

Evaluation time: 0.43
"Done","Done","Done","Done"
// Time 0.43
// Total time 0.43
// Using locale /usr/share/locale/
// ja_JP.UTF-8
// /usr/share/locale/
// giac
// UTF-8
// Maximum number of parallel threads 2
Added 0 synonyms
"Done","Done","Done","Done"
// Time 0
// Total time 0
[31,
 x^14+28*x^11+28*x^10-28*x^9+140*x^8+360*x^7+147*x^6+196*x^5+336*x^4-546*x^3
     -532*x^2+896*x+823,98,49,[7,7],5.794,25.028]
  
  *** nfisincl: Warning: increasing stack size to 16000000.
  *** write: Warning: increasing stack size to 16000000.
<permutation group with 75 generators>
// Using locale /usr/share/locale/
// ja_JP.UTF-8
// /usr/share/locale/
// giac
// UTF-8
// Maximum number of parallel threads 2
Added 0 synonyms

Evaluation time: 8.13
"Done","Done","Done","Done"
// Time 8.13
// Total time 8.13
// Using locale /usr/share/locale/
// ja_JP.UTF-8
// /usr/share/locale/
// giac
// UTF-8
// Maximum number of parallel threads 2
Added 0 synonyms
"Done","Done","Done","Done"
// Time 0.35
// Total time 0.35
// Using locale /usr/share/locale/
// ja_JP.UTF-8
// /usr/share/locale/
// giac
// UTF-8
// Maximum number of parallel threads 2
Added 0 synonyms
"Done","Done","Done","Done"
// Time 0.01
// Total time 0.01
[32,
 x^15-470*x^13-305*x^12+71840*x^11+85357*x^10-4292700*x^9-3714805*x^8
     +119761820*x^7+25284495*x^6-1542190154*x^5+717324725*x^4+7178878600*x^3
     -5452953875*x^2-7998223215*x+4461221029,75,75,[3,5,5],15.858,123.12]
  
Evaluation took 179.0620 seconds (218.4700 elapsed) using 74669.995 MB.