例題の多項式\( x^4+2\,x^3+3\,x^2+4\,x+5 \)の組成列は、
\(NSGS=\left[ Group\left \{1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 \right \} , Group\left \{1 , 4 , 5 , 8 , 9 , 12 , 13 , 16 , 17 , 20 , 21 , 24 \right \} , Group\left \{1 , 8 , 17 , 24 \right \} , Group\left \{1 , 8 \right \} , Group\left \{1 \right \} \right] \)
でした。合計5つの群からなっていますから、縮小／拡大は合計４ステップあります。前回の記事：
ではその第１ステップを実行し、体\(F_1\)に添加する元\(\alpha_{1}\)、\(\alpha_{1}\)を含む係数体\(F_2\)でのVの最小多項式g1を計算しました。
では今回は第２ステップであるNSGS[2]からNSGS[3]への群の縮小に基づく計算を行います。p1が3になる影響以外は前回と同じです。
(%i4) p1:NSGS[2]@degree/NSGS[3]@degree;
$$ \tag{%o4} 3 $$
p1=3なので1の３乗根を添加する必要があります。この時点でそれを行いました。
(%i5) if (p1>2) then (push([zeta[p1],ratsimp((zeta[p1]^p1-1)/(zeta[p1]-1))],C),C);
$$ \tag{%o5} \left[ \left[ \zeta_{3} , \zeta_{3}^2+\zeta_{3}+1 \right] , \left[ \alpha_{1} , \alpha_{1}^2-38880000 \right] \right] $$
(%i6) g1v:ev(g1,x:V);
$$ \tag{%o6} V^{12}+12\,V^{11}+96\,V^{10}+520\,V^9+2022\,V^8+5808\,V^7+\alpha_{1}\,\left(V^6+6\,V^5+24\,V^4+56\,V^3+\frac{33\,V^2}{5}-\frac{414\,V}{5}-\frac{82}{5}\right)+11420\,V^6+13704\,V^5+35793\,V^4+110564\,V^3+396300\,V^2+575760\,V+1352400 $$
(%i7) gr2:listify(NSGS[3]@index_element_set);
$$ \tag{%o7} \left[ 1 , 8 , 17 , 24 \right] $$
(%i8) h0:product(x-remainder(ev(V[gr2[i]],PS@vncond),g1v),i,1,length(gr2))$
(%i9) gr22:makelist(gr_mult(4,elem,NSGS[3]),elem,gr2);
$$ \tag{%o9} \left[ 4 , 12 , 13 , 21 \right] $$
(%i10) h1:product(x-remainder(ev(V[gr22[i]],PS@vncond),g1v),i,1,length(gr22))$
p1=3なのでh2まで計算する必要があります。そのために、gr22にもう一度"4"をかけてgr23を得ます。
(%i11) gr23:makelist(gr_mult(4,elem,NSGS[3]),elem,gr22);
$$ \tag{%o11} \left[ 5 , 9 , 16 , 20 \right] $$
(%i12) h2:product(x-remainder(ev(V[gr23[i]],PS@vncond),g1v),i,1,length(gr23))$
\(\theta_0\)を計算します。
(%i13) theta[0]:ef_polynomial_reduction(remainder((h0+h1+h2)/p1,g1v),C)$
ARRSTORE: use_fast_arrays=false; allocate a new property hash table for $THETA
\(\theta_{1}^3\)を計算します。
(%i14) theta[13]:ratexpand(ef_polynomial_reduction(
remainder(remainder((h0+zeta[p1]*h1+zeta[p1]^2*h2)/p1,g1v)^p1,g1v),C));
$$ \tag{%o14} -\frac{14\,\alpha_{1}\,\zeta_{3}\,x^6}{27}+1440\,\zeta_{3}\,x^6+
\frac{19\,\alpha_{1}\,x^6}{135}+2120\,x^6-\frac{28\,\alpha_{1}\,
\zeta_{3}\,x^5}{9}+8640\,\zeta_{3}\,x^5+\frac{38\,\alpha_{1}\,x^5
}{45}+12720\,x^5-\frac{692\,\alpha_{1}\,\zeta_{3}\,x^4}{45}+100800
\,\zeta_{3}\,x^4+\frac{14\,\alpha_{1}\,x^4}{45}+145020\,x^4-
\frac{5504\,\alpha_{1}\,\zeta_{3}\,x^3}{135}+345600\,\zeta_{3}\,x^
3-\frac{592\,\alpha_{1}\,x^3}{135}+495280\,x^3-\frac{3229\,
\alpha_{1}\,\zeta_{3}\,x^2}{90}+1018440\,\zeta_{3}\,x^2-\frac{
12247\,\alpha_{1}\,x^2}{180}+1748190\,x^2+\frac{67\,\alpha_{1}\,
\zeta_{3}\,x}{45}+1368720\,\zeta_{3}\,x-\frac{2251\,\alpha_{1}\,x
}{18}+2539740\,x+\frac{17927\,\alpha_{1}\,\zeta_{3}}{54}+375480\,
\zeta_{3}-\frac{113513\,\alpha_{1}}{540}+2793065 $$
\(A1, Q1, q1\)を計算します。
(%i15) A1:coeff(theta[13],x,hipow(theta[13],x));
$$ \tag{%o15} -\frac{14\,\alpha_{1}\,\zeta_{3}}{27}+1440\,\zeta_{3}+\frac{19\,\alpha_{1}}{135}+2120 $$
(%i16) Q1:ratexpand(ef_mult(theta[13],ef_divide(1,A1,C),C));
$$ \tag{%o16} x^6+6\,x^5+\frac{\alpha_{1}\,\zeta_{3}\,x^4}{312}-\frac{54\,\zeta_{3}\,x^4}{13}+\frac{19\,\alpha_{1}\,x^4}{15600}+\frac{831\,x^4}{26}+\frac{\alpha_{1}\,\zeta_{3}\,x^3}{78}-\frac{216\,\zeta_{3}\,x^3}{13}+\frac{19\,\alpha_{1}\,x^3}{3900}+\frac{1142\,x^3}{13}+\frac{2061\,\alpha_{1}\,\zeta_{3}\,x^2}{33800}-\frac{18522\,\zeta_{3}\,x^2}{169}+\frac{81\,\alpha_{1}\,x^2}{2704}+\frac{62871\,x^2}{676}+\frac{4883\,\alpha_{1}\,\zeta_{3}\,x}{50700}-\frac{31428\,\zeta_{3}\,x}{169}+\frac{5087\,\alpha_{1}\,x}{101400}+\frac{8895\,x}{338}+\frac{752281\,\alpha_{1}\,\zeta_{3}}{5272800}-\frac{1411803\,\zeta_{3}}{4394}+\frac{1189063\,\alpha_{1}}{10545600}-\frac{9640855}{17576} $$
(%i17) q1:ratexpand(ef_pthroot(Q1,p1,C));
$$ \tag{%o17} x^2+2\,x+\frac{\alpha_{1}\,\zeta_{3}}{936}-\frac{18\,\zeta_{3}}{13}+\frac{19\,\alpha_{1}}{46800}+\frac{173}{26} $$
添加する元は\(A1\)の３乗根ですが、これに\(\alpha_2\)と名前をつけます。
(%i18) a1:alpha[2];
$$ \tag{%o18} \alpha_{2} $$
\(\alpha_2\)を添加します。
(%i19) push([alpha[2],alpha[2]^p1-A1],C);
$$ \tag{%o19} \left[ \left[ \alpha_{2} , \frac{14\,\alpha_{1}\,\zeta_{3}}{27}-1440\,\zeta_{3}+\alpha_{2}^3-\frac{19\,\alpha_{1}}{135}-2120 \right] , \left[ \zeta_{3} , \zeta_{3}^2+\zeta_{3}+1 \right] , \left[ \alpha_{1} , \alpha_{1}^2-38880000 \right] \right] $$
\(\theta_1\)を計算します。
(%i20) theta[1]:a1*q1;
$$ \tag{%o20} \alpha_{2}\,\left(x^2+2\,x+\frac{\alpha_{1}\,\zeta_{3}}{936}-\frac{18\,\zeta_{3}}{13}+\frac{19\,\alpha_{1}}{46800}+\frac{173}{26}\right) $$
\(\theta_{2}^3\)を計算します。
(%i21) theta[23]:ratexpand(ef_polynomial_reduction(
remainder(remainder((h0+zeta[p1]^2*h1+zeta[p1]^4*h2)/p1,g1v)^p1,g1v),C));
$$ \tag{%o21} \frac{14\,\alpha_{1}\,\zeta_{3}\,x^6}{27}-1440\,\zeta_{3}\,x^6+
\frac{89\,\alpha_{1}\,x^6}{135}+680\,x^6+\frac{28\,\alpha_{1}\,
\zeta_{3}\,x^5}{9}-8640\,\zeta_{3}\,x^5+\frac{178\,\alpha_{1}\,x^5
}{45}+4080\,x^5+\frac{692\,\alpha_{1}\,\zeta_{3}\,x^4}{45}-100800
\,\zeta_{3}\,x^4+\frac{706\,\alpha_{1}\,x^4}{45}+44220\,x^4+
\frac{5504\,\alpha_{1}\,\zeta_{3}\,x^3}{135}-345600\,\zeta_{3}\,x^
3+\frac{4912\,\alpha_{1}\,x^3}{135}+149680\,x^3+\frac{3229\,
\alpha_{1}\,\zeta_{3}\,x^2}{90}-1018440\,\zeta_{3}\,x^2-\frac{5789
\,\alpha_{1}\,x^2}{180}+729750\,x^2-\frac{67\,\alpha_{1}\,\zeta_{3
}\,x}{45}-1368720\,\zeta_{3}\,x-\frac{11389\,\alpha_{1}\,x}{90}+
1171020\,x-\frac{17927\,\alpha_{1}\,\zeta_{3}}{54}-375480\,\zeta_{
3}-\frac{292783\,\alpha_{1}}{540}+2417585 $$
\(A2, Q2, q2\)を計算します。
(%i22) A2:coeff(theta[23],x,hipow(theta[23],x));
$$ \tag{%o22} \frac{14\,\alpha_{1}\,\zeta_{3}}{27}-1440\,\zeta_{3}+\frac{89\,\alpha_{1}}{135}+680 $$
(%i23) Q2:ratexpand(ef_mult(theta[23],ef_divide(1,A2,C),C));
$$ \tag{%o23} x^6+6\,x^5-\frac{\alpha_{1}\,\zeta_{3}\,x^4}{312}+\frac{54\,\zeta_{3}\,x^4}{13}-\frac{31\,\alpha_{1}\,x^4}{15600}+\frac{939\,x^4}{26}-\frac{\alpha_{1}\,\zeta_{3}\,x^3}{78}+\frac{216\,\zeta_{3}\,x^3}{13}-\frac{31\,\alpha_{1}\,x^3}{3900}+\frac{1358\,x^3}{13}-\frac{2061\,\alpha_{1}\,\zeta_{3}\,x^2}{33800}+\frac{18522\,\zeta_{3}\,x^2}{169}-\frac{2097\,\alpha_{1}\,x^2}{67600}+\frac{136959\,x^2}{676}-\frac{4883\,\alpha_{1}\,\zeta_{3}\,x}{50700}+\frac{31428\,\zeta_{3}\,x}{169}-\frac{4679\,\alpha_{1}\,x}{101400}+\frac{71751\,x}{338}-\frac{752281\,\alpha_{1}\,\zeta_{3}}{5272800}+\frac{1411803\,\zeta_{3}}{4394}-\frac{315499\,\alpha_{1}}{10545600}-\frac{3993643}{17576} $$
(%i24) q2:ratexpand(ef_pthroot(Q2,p1,C));
$$ \tag{%o24} x^2+2\,x-\frac{\alpha_{1}\,\zeta_{3}}{936}+\frac{18\,\zeta_{3}}{13}-\frac{31\,\alpha_{1}}{46800}+\frac{209}{26} $$
ちょっと技を使ってa2を計算します。この方法を使うとa2はすでに計算してCに添加済みの元\(\alpha_2\)で表せます。結果としてこれ以上元を添加する必要もありません。
(%i25) a2:alpha[2]^2*b[2],b[2]:(ef_mult(A1,A2,C)^(1/3))*ef_divide(1,A1,C);
$$ \tag{%o25} \frac{\alpha_{2}^2\,\left(\alpha_{1}\,\left(70\,\zeta_{3}+89\right)+5400\,\left(17-36\,\zeta_{3}\right)\right)}{9126000} $$
\(\theta_{2}\)を計算します。
(%i26) theta[2]:a2*q2;
$$ \tag{%o26} \frac{\alpha_{2}^2\,\left(\alpha_{1}\,\left(70\,\zeta_{3}+89\right)+5400\,\left(17-36\,\zeta_{3}\right)\right)\,\left(x^2+2\,x-\frac{\alpha_{1}\,\zeta_{3}}{936}+\frac{18\,\zeta_{3}}{13}-\frac{31\,\alpha_{1}}{46800}+\frac{209}{26}\right)}{9126000} $$
これでg2を計算できます。
(%i27) g2:ratexpand(ef_polynomial_reduction(theta[0]+theta[1]+theta[2],C));
$$ \tag{%o27} x^4+4\,x^3+\frac{7\,\alpha_{1}\,\alpha_{2}^2\,\zeta_{3}\,x^2}{912600}-\frac{18\,\alpha_{2}^2\,\zeta_{3}\,x^2}{845}+\frac{89\,\alpha_{1}\,\alpha_{2}^2\,x^2}{9126000}+\frac{17\,\alpha_{2}^2\,x^2}{1690}+\alpha_{2}\,x^2+16\,x^2+\frac{7\,\alpha_{1}\,\alpha_{2}^2\,\zeta_{3}\,x}{456300}-\frac{36\,\alpha_{2}^2\,\zeta_{3}\,x}{845}+\frac{89\,\alpha_{1}\,\alpha_{2}^2\,x}{4563000}+\frac{17\,\alpha_{2}^2\,x}{845}+2\,\alpha_{2}\,x+24\,x+\frac{103\,\alpha_{1}\,\alpha_{2}^2\,\zeta_{3}}{2281500}-\frac{348\,\alpha_{2}^2\,\zeta_{3}}{845}+\frac{\alpha_{1}\,\alpha_{2}\,\zeta_{3}}{936}-\frac{18\,\alpha_{2}\,\zeta_{3}}{13}+\frac{7\,\alpha_{1}\,\alpha_{2}^2}{182520}+\frac{601\,\alpha_{2}^2}{3380}+\frac{19\,\alpha_{1}\,\alpha_{2}}{46800}+\frac{173\,\alpha_{2}}{26}+230 $$
