冪根拡大の新たな構成法

\mathbb{Q} 上の既約多項式 f に対して,次が既知であるとします.
(1)f\mathbb{Q} 上の最小分解体 \operatorname{sp}(f) の定義多項式 f_\mathbb{Q}(つまり,\operatorname{sp}(f) の primitive element \alpha\mathbb{Q} 上の最小多項式
(2)f の各根 \in\mathbb{Q}[A]A\alpha に対応する変数)
(3)各 \sigma\in\operatorname{Gal}(\operatorname{sp}(f)/\mathbb{Q}) についての \sigma(A)(つまり, f_\mathbb{Q} の根 \in\mathbb{Q}[A])および f の根の列に対する置換としての \sigma の表示
(4)組成列 \operatorname{Gal}(\operatorname{sp}(f)/\mathbb{Q})=G_0\supset G_1\supset\ldots\supset G_n=\{\operatorname{id}\}
そして,p_i=|G_{i-1}|/|G_i|\,(i=1,\ldots,n) の全てが素数であるとします.

このとき,然るべき円分体 F_0 に幾つかの冪根 r_1,\ldots,r_m を逐次添加して,拡大体の列 \mathbb{Q}\subset F_0\subset F_1 \subset\ldots\subset F_m\,\left(\supset\operatorname{sp}(f)\right) を構成する,すなわち,
(5)F_0
および,各 i\in\{1,\ldots,m\} について
(6)F_i=F_{i-1}(r_i)F_{i-1} 上の定義多項式 f_i(つまり,r_iF_{i-1} 上の最小多項式
そして,各 i\in\{0,\ldots,m\} について
(7)\alphaF_i 上の最小多項式 g_i
を得る新たな方法を案出しましたので,その概要を述べます.


まず,|G_0| の各奇素因数 p に対する 1 の原始 p 乗根の 1\zeta_p\mathbb{Q} に添加した体を F_0 と定め,f_{\mathbb{Q}} の各 p 分体上の既約因数を 1 つずつ取り出し,それらの F_0 上での GCD を g_0 とする(各 p 分体上での因数分解は,その適当な部分体上での分解を挟むと処理時間が短縮できる場合がある).

次に,\{p_1,\ldots,p_n\},\ \{\zeta_{p_1},\ldots,\zeta_{p_n}\} の要素の総積をそれぞれ P,\ \zeta とおくと,各 i\in\{1,\ldots,n\} に対して,\zeta^{P/p_i}1 の原始 p_i 乗根となり,\tau\in G_{i-1}\setminus G_i を固定して


\begin{align}
M_i(X,j)&=\prod_{\sigma\in \tau^j G_i} (X-\sigma(A))\ \ (j=0,\ldots,p_i-1),\\
R_i(X,k)&=\sum_{j=0}^{p_i-1}  \zeta^{j k P/p_i}  M_i(X,j)\ \ (k=1,\ldots,p_i-1)
\end{align}

とおく.この右辺の計算は,g_{i-1},f_{i-1},\ldots,f_1i=1 のときは g_0のみ)および円分多項式 \Phi_{p_1},\ldots,\Phi_{p_n} による簡約を挟んで行い,

M_i(X,0)\in F_{i-1}[X] の場合,そこで処理を止め,F_i=F_{i-1},\ g_i=g_{i-1} と定めて,体の拡大は行わない(f_i は定義しない).

・そうでない場合,R_i(X,k) が零多項式にならない k があり,その零でない係数の 1 つを r_i とすると,{r_i}^j\ (j=1,\ldots,p_i-1) には A が含まれ,{r_i}^{p_i}\in F_{i-1} となるので,F_i=F_{i-1}(r_i),\ f_i(e_i)={e_i}^{p_i}-{r_i}^{p_i}e_ir_i に対応する変数だが,r_i の適当な有理数倍に対応させて余分な冪乗を含まない整数係数の monic な f_i を得ることもできる)と定める.また,

{e_i}^j-{r_i}^j=0\ \ (j=1,\ldots,p_i-1)

A^s\ (s=1,\ldots) についての連立 1 次方程式として解いたものを

\displaystyle M_i(X,0)=\prod_{\sigma\in G_i} (X-\sigma(A))

を簡約したものに含まれる各 A^s\ (s=1,\ldots) に代入して得られる多項式(この連立 1 次方程式の解は一般に一意的ではないが代入すれば打ち消し合う.結果の多項式は Groebner 基底や終結式から得ることもできる)を monic にしたものを g_i と定める.

以上で拡大しなかった体を取り除き,添え字を詰める.

上記の方法は(7)の算出において,冪根 r_ip_i-1 までの冪から得られる F_i 上の連立 1 次方程式を用いて A^s を直接消去する点が特徴で,全ての k に亘る Resolvent の和を取る方法があくまで多項式としての扱いを要請するのに対して,上記の R_i(X,k) は多くの場合に R_i(1,k) といったもの(つまり,\sigma(A)\ (\sigma\in G_i) の基本対称式の線形結合)で代替できます.

例1 f=x^3-3x-1 の場合
(1)f_{\mathbb{Q}}=A^3-3A-1
(3) [ [[1,2,3],A], [[2,3,1],2-A^2], [[3,1,2],A^2-A-2]]
(4) G_0\cong\{[1,2,3],[2,3,1],[3,1,2]\}G_1\cong\{[1,2,3]\}
(5)F_0=\mathbb{Q}(\zeta_3)\Phi_3=c_3^2+c_3+1c_p\zeta_p に対応する変数),g_0=A^3-3A-1
となり,\tau(A)=2-A^2 とすれば M_1(X,0)=X-A,\ M_1(X,1)=X+A^2-2,\ M_1(X,2)=X-A^2+A+2,従って, R_1(X,1)=(2c_3+1)A^2-(c_3+2)A-4c_3-2 なので,これ自体を r_1 として F_1=F_0(r_1),\ 3 e_1=r_1 とおけば,F_0 上では {r_1}^3=27c_3 により,f_1(e_1)=e_1^3-c_3 となり,3e_1={r_1},\ (3e_1)^2={r_1}^2 に対して A_s=A^s\ (s=1,2) とおいて得られる (-3 e_1)+A_2 (2c_3+1)-4c_3+A_1((-c_3)-2)-2=0,\ (-3e_1^2)+A_2(c_3+2)-2c_3+A_1((-2c_3)-1)-4=0F_1 上で解いた A_1 = (c_3+1)e_1^2-e_1,\ A_2 = e_1^2+((-c_3)-1)e_1+2X-A_1 に代入,変数 XA に改め(g_0 との F_1 上での GCD を求め)れば (-c_3e_1^2)-e_1^2+e_1+A となり,[[(-c_3 e_1^2)-e_1^2+e_1+A,A],[e_1^3-c_3,e_1],[c_3^2+c_3+1,c_3]] に至ります.

例2 実際には,連立方程式を解くことなく,A を主変数として e_i-r_i による剰余をとるだけで M_i(X,0) から A を消去できる場合が多く,以下,その場合には rem,連立方程式を解いた場合には ls を拡大次数 p_i と合わせて表示した実行例を掲載します.タイミングデータは,いずれも(1)から(7)までの算出時間の合計です.

[1] 
[2,rem] 
[x^2-2,[[e1+A,A],[e1^2-2,e1]]] 
[2] 
[3,ls] 
[x^3-3*x-1,[[(-c3*e1^2)-e1^2+e1+A,A],[e1^3-c3,e1],[c3^2+c3+1,c3]]] 
[3] 
[2,rem] 
[2,rem] 
[2,rem] 
[x^4-2,[[2*e3+e2+A,A],[e3^2+e1,e3],[e2^2-e1,e2],[e1^2-2,e1]]] 
[4] 
[2,rem] 
[2,rem] 
[2,rem] 
[x^4+x^2-1,[[(2*e3+e2)/2+A,A],[e3^2+2*e1+2,e3],[e2^2-2*e1+2,e2],[e1^2-5,e1]]] 
[5] 
[3,ls] 
[2,rem] 
[2,rem] 
[x^4-2*x^3+2*x^2+2,
 [[e3/21+A,A],[e3^2+42*e2+126*c3*e1^2+42*e1^2+294*e1+294,e3],
  [e2^2-210*e1^2+c3*((-42*e1^2)-588*e1)-294*e1+1323,e2],[e1^3+21*c3+14,e1],
  [c3^2+c3+1,c3]]]
  
[6] 
[2,rem] 
[3,ls] 
[2,rem] 
[2,rem] 
[x^4+2*x^3+3*x^2+4*x+5,
 [[(2*e4+e3-585)/585+A,A],
  [e4^2+c3*((-690*e1*e2^2)+315*e2^2+8775*e2)-30*e1*e2^2+675*e2^2+35100*e2
       +342225,e4],
  [e3^2+c3*(660*e1*e2^2+360*e2^2-35100*e2)+690*e1*e2^2-315*e2^2-26325*e2
       +342225,e3],[e2^3-8010*e1+c3*(4860-6300*e1)-2295,e2],[e1^2-3,e1],
  [c3^2+c3+1,c3]]]
  
[7] 
[2,rem] 
[3,ls] 
[2,rem] 
[2,rem] 
[x^4+x+1,
 [[(2*e4+e3)/624+A,A],
  [e4^2+c3*((-46*e1*e2^2)-126*e2^2+4992*e2)-2*e1*e2^2-270*e2^2+19968*e2,e4],
  [e3^2+c3*(44*e1*e2^2-144*e2^2-19968*e2)+46*e1*e2^2+126*e2^2-14976*e2,e3],
  [e2^3-1068*e1+c3*((-840*e1)-3888)+1836,e2],[e1^2-229,e1],[c3^2+c3+1,c3]]]
  
[8] 
using a factor (-30*c5^3)+10*c5^2-20*c5+A^5-10 over the cyclotomic_5 
[5,rem] 
[x^5-2,[[e3+A,A],[e3^5+30*c5^3-10*c5^2+20*c5+10,e3],[c5^4+c5^3+c5^2+c5+1,c5]]]
  
[9] 
[2,rem] 
[5,ls] 
[x^5-5*x+12,
 [[(c5^2*(e1*(21*e2^4+55*e2^3+75*e2^2)-90*e2^4+125*e2^3-250*e2^2)
    +c5^3*(e1*(21*e2^4+15*e2^3+75*e2^2)-20*e2^4+125*e2^3)-55*e2^4
    +e1*((-12*e2^4)+35*e2^3-25*e2^2)+c5*((-110*e2^4)+70*e1*e2^3-250*e2^2)
    -75*e2^3-125*e2^2+625*e2)
    /3125
    +A,A],
  [e2^5-1875*e1+c5^3*(3125-3750*e1)+c5^2*((-3750*e1)-9375)-6250*c5-3125,e2],
  [e1^2+10,e1],[c5^4+c5^3+c5^2+c5+1,c5]]]
  
[10] 
[2,rem] 
[5,ls] 
[x^5+20*x+32,
 [[A-(c5^2*(e1*(8*e2^4+40*e2^3+200*e2^2)+20*e2^4-50*e2^3+500*e2^2)
     +c5^3*(e1*(8*e2^4+20*e2^3+200*e2^2)+10*e2^4-50*e2^3)
     +c5*(30*e2^4+60*e1*e2^3+500*e2^2)+15*e2^4+e1*((-e2^4)+30*e2^3-150*e2^2)
     +250*e2^2-2500*e2)
     /12500,A],
  [e2^5+c5^2*(32500*e1+12500)+c5^3*(32500*e1-87500)+47500*e1-75000*c5-37500,
   e2],[e1^2+5,e1],[c5^4+c5^3+c5^2+c5+1,c5]]]
  
[11] 
[2,rem] 
[5,ls] 
[x^5+11*x+44,
 [[A-(c5^2*(e1*(189*e2^4+1045*e2^3+4235*e2^2)+350*e2^4-935*e2^3+6050*e2^2)
     +c5^3*(e1*(189*e2^4+385*e2^3+4235*e2^2)+100*e2^4-935*e2^3)
     +c5*(450*e2^4+1430*e1*e2^3+6050*e2^2)+225*e2^4
     +e1*((-83*e2^4)+715*e2^3-2420*e2^2)+385*e2^3+3025*e2^2-33275*e2)
     /166375,A],
  [e2^5+c5^2*(35475*e1-6875)+c5^3*(35475*e1-34375)+41800*e1-41250*c5-20625,
   e2],[e1^2+2,e1],[c5^4+c5^3+c5^2+c5+1,c5]]]
  
[12] 
[5,ls] 
[x^5+x^4-4*x^3-3*x^2+3*x+1,
 [[(c5^3*(10*e1^4-22*e1^3-363*e1^2)+16*e1^4+c5^2*((-10*e1^4)-99*e1^3-121*e1^2)
                                   +c5*((-25*e1^4)-66*e1^3+121*e1^2)-132*e1^3
                                   -121*e1^2+1331*e1+1331)
    /6655
    +A,A],[e1^5-165*c5^3-385*c5^2-275*c5-451,e1],[c5^4+c5^3+c5^2+c5+1,c5]]]
  
[13] 
using a factor 5000*c5^3+A^3*((-40*c5^3)-40*c5^2-20)+A^2*((-300*c5^3)+100*c5^2-200*c5-100)+7000*c5^2+12000*c5+A^5+1000*A+6000 over the cyclotomic_5
  
[5,ls] 
[x^5+100*x^2+1000,
 [[((-e3^4)+c5^2*((-2*e3^4)+6*e3^3-8*e3^2)+c5*((-2*e3^4)-12*e3^2)
           +c5^3*(6*e3^3-4*e3^2)-2*e3^3-6*e3^2+20*e3)
    /20
    +A,A],[e3^5+240*c5^3-80*c5^2+160*c5+80,e3],[c5^4+c5^3+c5^2+c5+1,c5]]]
  
[14] 
using a factor A^3-c3 over the cyclotomic_3 
[3,rem] 
[x^6+x^3+1,[[e2+A,A],[e2^3+c3,e2],[c3^2+c3+1,c3]]] 
[15] 
using a factor (-720*c3)+A^6-74 over the cyclotomic_3 
[2,rem] 
[3,rem] 
[x^6-2,[[e3+A,A],[e3^3-18*c3*e1-19*e1,e3],[e1^2-2,e1],[c3^2+c3+1,c3]]] 
[16] 
using a factor A^2*(1875*c5^3+1875*c5^2+3125)+A^6*(175*c5^3+175*c5^2+350)+5775*c5^3+A^8*((-10*c5^3)-10*c5^2-30)+A^4*((-1000*c5^3)-1000*c5^2-1750)+5775*c5^2+A^10+9450 over the cyclotomic_5
  
[2,rem] 
[5,ls] 
[x^5-5*x^3+5*x-5,
 [[A-(c5^2*(3*e2*e3^4-10*e3^4)+3*c5^3*e2*e3^4-2*e2*e3^4-10*c5*e3^4-5*e3^4
                              -80*e3)
     /160,A],[e3^5-80*e2-1200*c5^3+400*c5^2-800*c5-400,e3],
  [e2^2+231*c5^3+231*c5^2+378,e2],[c5^4+c5^3+c5^2+c5+1,c5]]]
  
[17] 
using a factor 162*c3+A^6*((-18*c3)-9)+A^9+108*A^3+81 over the cyclotomic_3 
[3,ls] 
[3,rem] 
[x^6+x^3+7,
 [[(7*e3+3*c3*e2^2+e2^2)/7+A,A],[e3^3+3*c3+1,e3],[e2^3-3*c3+5,e2],
  [c3^2+c3+1,c3]]]
  
[18] 
[2,rem] 
[3,ls] 
[2,rem] 
[2,rem] 
[x^6-3*x^4+1,
 [[e4+e3+A,A],
  [e4^2+c3*(4*e1*e2^2*e3-3*e2^2+e2)+e1*(4*e2^2*e3-4*e2*e3)-4*e2^2+4*e2-5,e4],
  [e3^2+e2^2-c3*e2-e2-1,e3],[e2^3-c3,e2],[e1^2+1,e1],[c3^2+c3+1,c3]]]
  
[19] 
[2,rem] 
[3,ls] 
[2,rem] 
[2,rem] 
[x^6+x^4-9,
 [[e4/156+A,A],
  [e4^2+1872*e3+c3*((-3780*e1*e2^2)-52056*e2^2)+1026*e1*e2^2-76638*e2^2
       +4056*e2+40560,e4],
  [e3^2+c3*(e1*(2*e2^2-260*e2)+810*e2^2+4212*e2)+432*e2^2
       +e1*((-44*e2^2)-364*e2)-1404*e2,e3],
  [e2^3-3204*e1+c3*((-2520*e1)-34704)+16388,e2],[e1^2+239,e1],[c3^2+c3+1,c3]]]
  
[20] 
[2,rem] 
[2,rem] 
[3,ls] 
[2,rem] 
[2,rem] 
[x^6+x^4-8,
 [[(e5+14*e4)/1029+A,A],
  [e5^2+c3*(e1*(e2*(294*e3*e4-282*e3^2*e4)+583884*e3^2)
           +e2*(864*e3^2*e4+21168*e3*e4)+4655784*e3^2-230496*e3)
       +e1*(e2*(1911*e3*e4-69*e3^2*e4)+683550*e3^2)
       +e2*(2970*e3^2*e4+7938*e3*e4)-2878407*e3^2+266511*e3+3529470,e5],
  [e4^2+c3*((-1692*e1*e3^2)+5136*e3^2+1176*e3)-414*e1*e3^2+17655*e3^2+441*e3
       +7203,e4],[e3^3+c3*(1716*e1+38520)-2382*e1+34561,e3],[e2^2-2,e2],
  [e1^2+106,e1],[c3^2+c3+1,c3]]]
  
[21] 
<permutation group with 72 generators>
[2,rem] 
[2,rem] 
[2,rem] 
[3,ls] 
[3,ls] 
[x^6+x^4-x^2+5*x-5,
 [[A-(c3*(e1*(20*e3*e5^2-486*e5^2-7938*e4^2)-60*e3*e5^2+810*e5^2-13230*e4^2)
     +e1*(37*e3*e5^2+27*e5^2-441*e2*e4^2-3969*e4^2)-111*e3*e5^2-45*e5^2
     -1176*e5-1323*e2*e4^2-6615*e4^2-3528*e4)
     /7056,A],[e5^3+c3*(240*e3+1944*e1)-204*e3+2052*e1,e5],
  [e4^3-4*e2+24*c3*e1+12*e1,e4],[e3^2+4*e1+143,e3],[e2^2-4*e1+143,e2],
  [e1^2-5,e1],[c3^2+c3+1,c3]]]
  
[22] 
using a factor 42*c7^4+A^2*((-14*c7^4)-14*c7^2-14*c7-7)+42*c7^2+42*c7+A^7-7*A^5+49*A+21 over the cyclotomic_7
  
[7,ls] 
[x^7+7*x^3+7*x^2+7*x-1,
 [[(c7^4*(25*e2^6+56*e2^5-490*e2^4+686*e2^3-2401*e2^2)
    +c7*(24*e2^6-490*e2^4-4802*e2^2)+c7^5*(16*e2^6-28*e2^5-343*e2^4-4802*e2^2)
    +c7^2*(8*e2^6-28*e2^5-147*e2^4)+12*e2^6
    +c7^3*((-e2^6)+56*e2^5+686*e2^3-2401*e2^2)+91*e2^5-245*e2^4+1029*e2^3
    -2401*e2^2+16807*e2)
    /117649
    +A,A],
  [e2^7+84035*c7^5-184877*c7^4-689087*c7^3-957999*c7^2-873964*c7-436982,e2],
  [c7^6+c7^5+c7^4+c7^3+c7^2+c7+1,c7]]]
  
[23] 
using a factor A*(392*c7^5+1176*c7^4+1176*c7^3+392*c7^2-784)+84*c7^5+A^3*((-56*c7^5)-280*c7^4-280*c7^3-56*c7^2+280)+A^5*(14*c7^4+14*c7^3-28)+224*c7^4+224*c7^3+84*c7^2+A^7-126 over the cyclotomic_21
  
[7,ls] 
[x^7-14*x^5+56*x^3-56*x+22,
 [[A-(37*c7^5*e2^6+34*c7^4*e2^6+c7^3*e2^6+15*c7^2*e2^6+44*c7*e2^6+16*e2^6
                  -224*e2)
     /224,A],[e2^7-196*c7^5+42*c7^4+196*c7^3+574*c7^2+462*c7+294,e2],
  [c7^6+c7^5+c7^4+c7^3+c7^2+c7+1,c7]]]
  
[24] 
using a factor (-84*c7^5)-112*c7^4-28*c7^3-56*c7^2-140*c7+A^7-70 over the cyclotomic_21
  
[7,rem] 
[x^7-2,
 [[e3+A,A],[e3^7+84*c7^5+112*c7^4+28*c7^3+56*c7^2+140*c7+70,e3],
  [c7^6+c7^5+c7^4+c7^3+c7^2+c7+1,c7]]]
  
[25] 
[2,rem] 
[2,rem] 
[2,rem] 
[2,rem] 
[x^8-2*x^6-x^4+7*x^2-5*x+1,
 [[(e4+e3+2*e2+4*e1)/8+A,A],[e4^2+e1*(6*e2+8)-2*e2-32,e4],
  [e3^2+e1*(2*e2-8)-14*e2-32,e3],[e2^2+2*e1+10,e2],[e1^2-5,e1]]]
  
[26] 
[2,rem] 
[3,ls] 
[3,ls] 
[x^9+x^8+3*x^6+3*x^3-3*x^2+5*x-1,
 [[(c3*(e2^2*(680352*e3^2+180787068)+808082*e1*e2*e3^2+202593204*e3^2)
    +e1*(404041*e2*e3^2-12575046*e3^2+e2^2*((-40541*e3^2)-116220258))
    +e2*(9707841*e3^2+18750201624)+e2^2*(340176*e3^2+90393534)+101296602*e3^2
    +1704563784*e3)
    /337503629232
    +A,A],
  [e3^3-35079*e2^2+e1*((-6237*e2^2)+32670*e2-574992)
       +c3*((-70158*e2^2)+65340*e1*e2-13224816)-498762*e2-6612408,e3],
  [e2^3+108*e1+168*c3+84,e2],[e1^2+199,e1],[c3^2+c3+1,c3]]]
  
[27] 
using a factor A^6*((-104*c3)-52)+A^12-3 over the cyclotomic_3 
[2,rem] 
[2,rem] 
[3,rem] 
[x^12-3,
 [[e4/6+A,A],[e4^3-72*c3*e1*e2-36*e1*e2-108*e2,e4],[e2^2-52*e1-90,e2],
  [e1^2-3,e1],[c3^2+c3+1,c3]]]
  
[28] 
using a factor A*(392*c7^5+1176*c7^4+1176*c7^3+392*c7^2-784)+84*c7^5+A^3*((-56*c7^5)-280*c7^4-280*c7^3-56*c7^2+280)+A^5*(14*c7^4+14*c7^3-28)+224*c7^4+224*c7^3+84*c7^2+A^7-126 over the cyclotomic_21
  
[7,ls] 
[x^7-14*x^5+56*x^3-56*x+22,
 [[A-(37*c7^5*e2^6+34*c7^4*e2^6+c7^3*e2^6+15*c7^2*e2^6+44*c7*e2^6+16*e2^6
                  -224*e2)
     /224,A],[e2^7-196*c7^5+42*c7^4+196*c7^3+574*c7^2+462*c7+294,e2],
  [c7^6+c7^5+c7^4+c7^3+c7^2+c7+1,c7]]]
  
[29] 
[3,ls] 
[5,ls] 
[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,
 [[(c5^2*(c3*(e1*(525953392818*e2^4+46208521901700*e2^3+5483213506110420*e2^2)
             +e1^2*((-62302134318*e2^4)-29074162205952*e2^3
                                       -610836365423592*e2^2))
         +e1*(116400133372*e2^4-111709403146002*e2^3+1413356700403458*e2^2)
         +e1^2*(35740453538*e2^4-16527048188010*e2^3+404838613165410*e2^2)
         +2554940802905*e2^4+5884406978016*e2^3+298462719049948368*e2^2)
    +c5*(c3*(e1^2*(79051166520*e2^4+13259732939685*e2^3+1650568846264596*e2^2)
            +e1*((-13970179140*e2^4)-61606115074863*e2^3
                                    +8506698305265252*e2^2))
        +e1*(396789211070*e2^4+17170190959320*e2^3+15616854293421456*e2^2)
        +e1^2*(63547608910*e2^4+782091603927*e2^3+2793257089431372*e2^2)
        -1132510352465*e2^4-694234588400388*e2^3+121491722900512989*e2^2)
    +c5^3*(c3*(e1^2*(60994007362*e2^4+9958452196578*e2^3
                                     -1843804912740354*e2^2)
              +e1*((-8979432782*e2^4)-26288677701354*e2^3
                                     +1084484608686126*e2^2))
          +e1*(307652844052*e2^4+29544771597858*e2^3-8622588208586724*e2^2)
          +e1^2*(49331767338*e2^4+3917263880256*e2^3-1355756659169274*e2^2)
          -2420808895895*e2^4-1081854379975083*e2^3+173521417632078870*e2^2)
    +c3*(e1*(338714791972*e2^4+103803801230007*e2^3+5905815006243258*e2^2)
        +e1^2*((-8749508036*e2^4)-8422621998885*e2^3-5198921087024934*e2^2
                                 +359767749025048144986))
    +e1*(237056535124*e2^4+42986287364100*e2^3-21939579777759444*e2^2
                          +1858800036629415415761)
    +e1^2*(49161208632*e2^4+10281781872597*e2^3-3348131738146902*e2^2
                           +299806457520873454155)-116094621149*e2^4
    -95925022751778*e2^3+270729765065001618*e2^2+59961291504174690831*e2
    -1858800036629415415761)
    /27882000549441231236415
    +A,A],
  [e2^5+c5*(6358442085*e1^2+c3*(313059766185*e1-54981822735*e1^2)
                           -23189612310*e1-2516072935635)-101660268159*e1^2
       +c5^2*((-13838962185*e1^2)+c3*(150732480015*e1-46753250625*e1^2)
                                 -115948061550*e1-2156633944830)
       +c3*((-103904424189*e1^2)-90439488009*e1)
       +c5^3*((-116696113560*e1^2)+c3*((-89018189190*e1^2)-255085735410*e1)
                                  -672498756990*e1-3594389908050)
       -612205764984*e1-3881941100694,e2],[e1^3+186*c3+31,e1],[c3^2+c3+1,c3],
  [c5^4+c5^3+c5^2+c5+1,c5]]]
  
Evaluation took 42.7050 seconds (48.6050 elapsed) using 25825.972 MB.

例3

(%i3) for i:1 thru 35 do (print([i]),p:x^i-2,RR7r_gp(cs_gap(spnGG_gp(p))))$
[1] 
Group(())
[2] 
Group([ (), (1,2) ])
[2,rem] 
[3] 
Group([ (), (2,3), (1,3,2), (1,2), (1,3), (1,2,3) ])
using a factor (-12*c3)+A^3-6 over the cyclotomic_3 
[3,rem] 
[4] 
Group([ (), (1,4), (2,3), (1,4)(2,3), (1,3)(2,4), (1,3,4,2), (1,2,4,3), (1,2)(3,4) ])
[2,rem] 
[2,rem] 
[2,rem] 
[5] 
<permutation group with 20 generators>
using a factor (-30*c5^3)+10*c5^2-20*c5+A^5-10 over the cyclotomic_5 
[5,rem] 
[6] 
Group([ (), (1,5)(2,6), (1,2)(3,4)(5,6), (1,6)(2,5)(3,4), (1,6)(2,4)(3,5), (1,2,4,6,5,3), (1,5,4)(2,3,6), (2,3)(4,5), (1,3,5,6,4,2), (1,3)(2,5)(4,6), (1,4)(3,6), (1,4,5)(2,6,3) ])
using a factor (-720*c3)+A^6-74 over the cyclotomic_3 
[2,rem] 
[3,rem] 
[7] 
<permutation group with 42 generators>
using a factor (-84*c7^5)-112*c7^4-28*c7^3-56*c7^2-140*c7+A^7-70 over the cyclotomic_21
  
[7,rem] 
[8] 
<permutation group with 16 generators>
[2,rem] 
[2,rem] 
[2,rem] 
[2,rem] 
[9] 
<permutation group with 54 generators>
using a factor 1296*c3+A^18*((-1008*c3)-504)+A^27+17172*A^9+648 over the cyclotomic_3
  
[3,ls] 
[3,rem] 
[3,rem] 
[10] 
<permutation group with 40 generators>
using a factor (-26840*c5^3)-57200*c5^2-51480*c5+A^10-16282 over the cyclotomic_5
  
[2,rem] 
[5,rem] 
[11] 
<permutation group with 110 generators>
using a factor (-1254*c11^9)-220*c11^8-308*c11^7-990*c11^6+330*c11^5-352*c11^4-440*c11^3+594*c11^2-660*c11+A^11-330 over the cyclotomic_55
  
[11,rem] 
[12] 
<permutation group with 48 generators>
using a factor (-5100480*c3)+A^12*((-744480*c3)-18500)+A^24-21345404 over the cyclotomic_3
  
[2,rem] 
[2,rem] 
[2,rem] 
[3,rem] 
[13] 
<permutation group with 156 generators>
using a factor (-3146*c13^11)-2730*c13^10+858*c13^9-2600*c13^8-4004*c13^7-1144*c13^6-2548*c13^5-6006*c13^4-2418*c13^3-2002*c13^2-5148*c13+A^13-2574 over the cyclotomic_39
  
[13,rem] 
[14] 
<permutation group with 84 generators>
using a factor (-2996392*c7^5)-2076256*c7^4+590408*c7^3-1613920*c7^2-3503136*c7+A^14-613090 over the cyclotomic_21
  
[2,rem] 
[7,rem] 
[15] 
<permutation group with 120 generators>
using a factor ((-15780*c3)-26520)*c5^3+((-15780*c3)-3740)*c5^2-14480*c5-25092*c3+A^15-19786 over the cyclotomic_15
  
[3,rem] 
[5,rem] 
[16] 
<permutation group with 64 generators>
[2,rem] 
[2,rem] 
[2,rem] 
[2,rem] 
[2,rem] 
[2,rem] 
[17] 
  *** nfisincl: Warning: increasing stack size to 16000000.
<permutation group with 272 generators>
using a factor (-51272*c17^15)-12104*c17^14-11016*c17^13-60996*c17^12-7616*c17^11-12342*c17^10-37128*c17^9+12376*c17^8-12410*c17^7-17136*c17^6+36244*c17^5-13736*c17^4-12648*c17^3+26520*c17^2-24752*c17+A^17-12376 over the cyclotomic_17
  
[17,rem] 
[18] 
<permutation group with 108 generators>
using a factor (-525994305879853440*c3)+A^36*((-609493248*c3)-5515782)+A^18*(349624993158156-362748920014656*c3)+A^54-143361159962807816 over the cyclotomic_3
  
[2,rem] 
[3,ls] 
[3,rem] 
[3,rem] 
[19] 
  *** nfsplitting: Warning: increasing stack size to 16000000.
  *** nfisincl: Warning: increasing stack size to 32000000.
<permutation group with 342 generators>
using a factor (-124032*c19^17)-101118*c19^16+83980*c19^15-102714*c19^14-108528*c19^13+50388*c19^12-100814*c19^11-155040*c19^10-46512*c19^9-100738*c19^8-251940*c19^7-93024*c19^6-98838*c19^5-285532*c19^4-100434*c19^3-77520*c19^2-201552*c19+A^19-100776 over the cyclotomic_57
  
[19,rem] 
[20] 
<permutation group with 160 generators>
using a factor 9284352000000*c5^3+A^20*((-1330401280*c5^3)+2469629120*c5^2-2134125600*c5+682330108)+23014398000000*c5^2+22270413000000*c5+A^40+7977616362500 over the cyclotomic_5
  
[2,rem] 
[2,rem] 
[2,rem] 
[5,rem] 
[21] 
  *** nfisincl: Warning: increasing stack size to 16000000.
<permutation group with 252 generators>
using a factor (677124*c3-619248)*c7^5+((-447300*c3)-1144066)*c7^4+((-447300*c3)-520072)*c7^3+(677124*c3+79534)*c7^2-1216838*c7-505398*c3+A^21-861118 over the cyclotomic_21
  
[3,rem] 
[7,rem] 
[22] 
  *** nfsplitting: Warning: increasing stack size to 16000000.
<permutation group with 220 generators>
using a factor (-11229074560*c11^9)+6698642632*c11^8-2683141120*c11^7-5242592256*c11^6+9725452704*c11^5-6748408744*c11^4-1426168832*c11^3+5442664832*c11^2-11929283520*c11+A^22+1429976062 over the cyclotomic_55
  
[2,rem] 
[11,rem] 
[23] 
  *** nfsplitting: Warning: increasing stack size to 16000000.
  *** nfsplitting: Warning: increasing stack size to 32000000.
  *** nfisincl: Warning: increasing stack size to 64000000.
  *** _*_: Warning: increasing stack size to 16000000.
<permutation group with 506 generators>
using a factor (-2124694*c23^21)-489808*c23^20-423016*c23^19-3194470*c23^18-472604*c23^17-486772*c23^16-2778446*c23^15-288420*c23^14-490268*c23^13-1470942*c23^12+490314*c23^11-490360*c23^10-692208*c23^9+1797818*c23^8-493856*c23^7-508024*c23^6+2213842*c23^5-557612*c23^4-490820*c23^3+1144066*c23^2-980628*c23+A^23-490314 over the cyclotomic_253
  
[23,rem] 
[24] 
<permutation group with 96 generators>
using a factor A^36*((-6432*c3)-3216)+A^12*((-32604864*c3)-16302432)+A^48-1797988*A^24+4 over the cyclotomic_3
  
[2,rem] 
[2,rem] 
[2,rem] 
[2,rem] 
[3,rem] 
[25] 
  *** nfsplitting: Warning: increasing stack size to 16000000.
  *** nfsplitting: Warning: increasing stack size to 32000000.
  *** nfisincl: Warning: increasing stack size to 64000000.
  *** _*_: Warning: increasing stack size to 16000000.
<permutation group with 500 generators>
using a factor A^50*(3628746377641385000*c5^3-1647791371390805000*c5^2+1980955006250580000*c5+990477503125290000)-188500000*c5^3+A^100*((-33218900*c5^3)+32156300*c5^2-1062600*c5-531300)+A^75*((-75134626573500*c5^3)-75134626573500*c5^2+115030173419500)+A^25*((-427357389517500000*c5^3)-427357389517500000*c5^2+269980735334750000)+44500000*c5^2-144000000*c5+A^125-72000000 over the cyclotomic_5
  
[5,ls] 
[5,rem] 
[5,rem] 
[26] 
  *** nfsplitting: Warning: increasing stack size to 16000000.
  *** nfsplitting: Warning: increasing stack size to 32000000.
<permutation group with 312 generators>
using a factor (-971321120640*c13^11)-254360271360*c13^10+414622391040*c13^9-591564945816*c13^8-397311582720*c13^7+543980124480*c13^6-352472480256*c13^5-757760340480*c13^4+231240186840*c13^3-242359104000*c13^2-1094179257600*c13+A^26-104832219650 over the cyclotomic_39
  
[2,rem] 
[13,rem] 
[27] 
  *** nfsplitting: Warning: increasing stack size to 16000000.
  *** nfsplitting: Warning: increasing stack size to 32000000.
  *** nfisincl: Warning: increasing stack size to 64000000.
  *** _*_: Warning: increasing stack size to 16000000.
<permutation group with 486 generators>
using a factor A^54*(64185079264587155422066316743920384*c3+32092539632293577711033158371960192)+A^162*(646330905226424471479920*c3+323165452613212235739960)-1632586752*c3+A^216*((-168725700*c3)-84362850)+A^108*((-23162703161108643394416419869245312*c3)-11581351580554321697208209934622656)+A^243+26113852523044440*A^189+431757909098770766278078746432*A^135+515882115360428367277687574283383232*A^81+93715791643684117316945664*A^27-816293376 over the cyclotomic_3
  
[3,ls] 
[3,ls] 
[3,ls] 
[3,rem] 
[3,rem] 
[28] 
  *** nfsplitting: Warning: increasing stack size to 16000000.
  *** nfsplitting: Warning: increasing stack size to 32000000.
<permutation group with 336 generators>
using a factor 8310128683316942272*c7^5+A^28*((-14292346192992*c7^5)-7637767388160*c7^4+7443111224640*c7^3-6714427087872*c7^2-19353706444800*c7-3689457900548)-3955704597347134464*c7^4+9824604059141698176*c7^3-2664151871716256768*c7^2+6167144382212136960*c7+A^56+2799751623397086212 over the cyclotomic_21
  
[2,rem] 
[2,rem] 
[2,rem] 
[7,rem] 
[29] 
  *** nfsplitting: Warning: increasing stack size to 16000000.
  *** nfsplitting: Warning: increasing stack size to 32000000.
  *** nfsplitting: Warning: increasing stack size to 64000000.
  *** nfisincl: Warning: increasing stack size to 128000000.
  *** subst: Warning: increasing stack size to 16000000.
  *** subst: Warning: increasing stack size to 32000000.
  *** _*_: Warning: increasing stack size to 64000000.
<permutation group with 812 generators>
  *** nffactor: Warning: increasing stack size to 16000000.
using a factor (-89224590*c29^27)-20029198*c29^26-16908450*c29^25-155757840*c29^24-19982508*c29^23-19792500*c29^22-175147530*c29^21-19079970*c29^20-20022702*c29^19-123821880*c29^18-11445720*c29^17-20029952*c29^16-60090030*c29^15+20030010*c29^14-20030068*c29^13-28614300*c29^12+83761860*c29^11-20037318*c29^10-20980050*c29^9+135087510*c29^8-20267520*c29^7-20077512*c29^6+115697820*c29^5-23151570*c29^4-20030822*c29^3+49164570*c29^2-40060020*c29+A^29-20030010 over the cyclotomic_203
  
[29,rem] 
[30] 
  *** nfsplitting: Warning: increasing stack size to 16000000.
<permutation group with 240 generators>
using a factor ((-84445822933920*c3)-80494119004000)*c5^3+(9957570973280-34779365611320*c3)*c5^2+((-37324603531560*c3)-78290214838440)*c5-105867478586400*c3+A^30-37885424529826 over the cyclotomic_15
  
[2,rem] 
[3,rem] 
[5,rem] 
[31] 
  *** nfsplitting: Warning: increasing stack size to 16000000.
  *** nfsplitting: Warning: increasing stack size to 32000000.
  *** nfsplitting: Warning: increasing stack size to 64000000.
  *** nfsplitting: Warning: increasing stack size to 128000000.
  *** nfisincl: Warning: increasing stack size to 256000000.
  *** subst: Warning: increasing stack size to 16000000.
  *** subst: Warning: increasing stack size to 32000000.
  *** _*_: Warning: increasing stack size to 64000000.
<permutation group with 930 generators>
  *** nffactor: Warning: increasing stack size to 16000000.
  *** nffactor: Warning: increasing stack size to 16000000.
  *** nffactor: Warning: increasing stack size to 16000000.
  *** nffactor: Warning: increasing stack size to 32000000.
  *** nffactor: Warning: increasing stack size to 64000000.
  *** nffactor: Warning: increasing stack size to 128000000.
  *** nffactor: Warning: increasing stack size to 256000000.
using a factor (-209664780*c31^29)-169345560*c31^28+243161520*c31^27-174603780*c31^26-169407560*c31^25+431735760*c31^24-169684452*c31^23-170817192*c31^22+361020420*c31^21-169353620*c31^20-185122080*c31^19+112896420*c31^18-169344692*c31^17-258048960*c31^16-80640300*c31^15-169344568*c31^14-451585680*c31^13-153567180*c31^12-169335640*c31^11-699709680*c31^10-167872068*c31^9-169004808*c31^8-770425020*c31^7-169281700*c31^6-164085480*c31^5-581850780*c31^4-169343700*c31^3-129024480*c31^2-338689260*c31+A^31-169344630 over the cyclotomic_465
  
[31,rem] 
[32] 
  *** nfsplitting: Warning: increasing stack size to 16000000.
<permutation group with 256 generators>
[2,rem] 
[2,rem] 
[2,rem] 
[2,rem] 
[2,rem] 
[2,rem] 
[2,rem] 
[2,rem] 
[33] 
  *** nfsplitting: Warning: increasing stack size to 16000000.
  *** nfsplitting: Warning: increasing stack size to 32000000.
  *** nfsplitting: Warning: increasing stack size to 64000000.
  *** nfisincl: Warning: increasing stack size to 128000000.
  *** subst: Warning: increasing stack size to 16000000.
  *** _*_: Warning: increasing stack size to 32000000.
<permutation group with 660 generators>
using a factor c11^7*(519052776*c3+414162232)+c11^4*(519052776*c3-1195709680)-1920480816*c3+c11^2*(859393656-1137871680*c3)+c11^9*((-1137871680*c3)-3297865560)+c11^5*((-1331455950*c3)-513832374)+c11^6*((-1331455950*c3)-2118223800)+c11^8*((-3480414828*c3)-1221726616)+c11^3*((-3480414828*c3)-3559288436)-1300600224*c11+A^33-1610540520 over the cyclotomic_165
  
[3,rem] 
[11,rem] 
[34] 
  *** nfsplitting: Warning: increasing stack size to 16000000.
  *** nfsplitting: Warning: increasing stack size to 32000000.
  *** nfsplitting: Warning: increasing stack size to 64000000.
  *** nfsplitting: Warning: increasing stack size to 128000000.
  *** subst: Warning: increasing stack size to 16000000.
  *** _*_: Warning: increasing stack size to 32000000.
<permutation group with 544 generators>
using a factor (-3808228426557440*c17^15)-7919863789291520*c17^14-3403121175472128*c17^13+1279988834931712*c17^12-3539846083028992*c17^11-8119861978524672*c17^10-3604322525292544*c17^9+372311085462272*c17^8-3225651920873472*c17^7-6439119406838272*c17^6-4240802890164224*c17^5-1573811911309704*c17^4-2076123785922560*c17^3-4675445439768440*c17^2-5956765649455104*c17+A^34-2908228400359426 over the cyclotomic_17
  
[2,rem] 
[17,rem] 
[35] 
  *** nfsplitting: Warning: increasing stack size to 16000000.
  *** nfsplitting: Warning: increasing stack size to 32000000.
  *** nfsplitting: Warning: increasing stack size to 64000000.
  *** nfsplitting: Warning: increasing stack size to 128000000.
  *** subst: Warning: increasing stack size to 16000000.
  *** subst: Warning: increasing stack size to 32000000.
  *** _*_: Warning: increasing stack size to 64000000.
<permutation group with 840 generators>
using a factor ((-2814705810*c5^3)-11072532380*c5^2-1236699170*c5-9449209994)*c7^5+((-12865509020*c5^3)-12027706250*c5^2-11980726170*c5-18890856012)*c7^4+((-2956013550*c5^3)-2118210780*c5^2-11980726170*c5-9081390038)*c7^3+(6832839580*c5^3-1424986990*c5^2-1236699170*c5-7779009056)*c7^2+((-3002993630*c5^3)-3002993630*c5^2-15991519880)*c7-7609289760*c5^3+1673794090*c5^2-2932502040*c5+A^35-9462010960 over the cyclotomic_105
  
[5,rem] 
[7,rem] 
Evaluation took 467.7820 seconds (6274.2230 elapsed) using 340797.419 MB.