这个五次方程怎么解？

nyy

nyy 发表于 2024-1-15 14:08
f=32*x^5+3349456*x^4-5941616812296*x^3-585145514845851080*x^2+147013447513276833423286*x+1537730 ...

1. Clear["Global`*"];(*Clear all variables*)
   
2. (*定义多项式,这个多项式的伽罗瓦群是可解的*)
   
3. f=32*x^5+3349456*x^4-5941616812296*x^3-585145514845851080*x^2+147013447513276833423286*x+15377302441624829616294559439;
   
4. aa={Factor[f,Modulus->#],#}&/@Prime@Range[100];
   
5. MatrixForm[aa]
   

复制代码

计算结果
\[\left(
\begin{array}{cc}
1 & 2 \\
2 \left(x^5+2 x^4+2 x^2+2 x+1\right) & 3 \\
2 \left(x^5+3 x^4+2 x^3+3 x+2\right) & 5 \\
4 \left(x^5+3 x^4+6 x^3+3 x^2+4 x+3\right) & 7 \\
10 x^5 & 11 \\
6 \left(x^5+x^4+5 x^3+8 x^2+11 x+5\right) & 13 \\
15 \left(x^5+10 x^4+13 x^3+3 x^2+8\right) & 17 \\
13 \left(x^5+9 x^4+17 x^3+5 x^2+x+11\right) & 19 \\
9 x^4 (x+9) & 23 \\
3 \left(x^5+24 x^4+19 x^3+20 x^2+10 x+24\right) & 29 \\
x^5+30 x^4+26 x^3+13 x^2+x+7 & 31 \\
32 \left(x^5+16 x^4+16 x^3+12 x^2+34 x+11\right) & 37 \\
32 \left(x^5+18 x^4+17 x^3+27 x^2+3 x+31\right) & 41 \\
32 (x+13) (x+16) (x+25) (x+28) (x+34) & 43 \\
32 \left(x^5+25 x^4+22 x^3+34 x^2+27 x+2\right) & 47 \\
32 \left(x^5+22 x^4+29 x^3+44 x^2+x+42\right) & 53 \\
32 \left(x^5+34 x^4+37 x^3+49 x^2+24 x+37\right) & 59 \\
32 \left(x^5+25 x^4+23 x^3+52 x^2+38 x+39\right) & 61 \\
32 (x+29) (x+46) (x+47) (x+63) (x+66) & 67 \\
32 \left(x^5+52 x^4+64 x^3+27 x^2+5 x+19\right) & 71 \\
32 \left(x^5+25 x^4+38 x^3+62 x^2+8 x+28\right) & 73 \\
32 \left(x^5+35 x^4+49 x^3+77 x^2+71 x+64\right) & 79 \\
32 \left(x^5+49 x^4+4 x^3+68 x^2+50 x+79\right) & 83 \\
32 (x+3) (x+34) (x+40) (x+72) (x+80) & 89 \\
32 \left(x^5+56 x^4+18 x^3+56 x^2+77 x+54\right) & 97 \\
32 \left(x^5+85 x^4+80 x^3+55 x^2+64 x+23\right) & 101 \\
32 \left(x^5+74 x^4+2 x^3+75 x^2+64 x+37\right) & 103 \\
32 \left(x^5+78 x^4+84 x^3+53 x^2+10 x+34\right) & 107 \\
32 (x+67) (x+69) (x+78) (x+95) (x+103) & 109 \\
32 \left(x^5+89 x^4+103 x^3+83 x+32\right) & 113 \\
32 \left(x^5+86 x^4+108 x^3+12 x^2+48 x+7\right) & 127 \\
32 (x+49) (x+69) (x+101) (x+112) (x+129) & 131 \\
32 \left(x^5+71 x^4+72 x^3+22 x^2+13 x+112\right) & 137 \\
32 \left(x^5+73 x^4+14 x^3+35 x^2+66 x+103\right) & 139 \\
32 \left(x^5+147 x^4+87 x^3+66 x^2+113 x+87\right) & 149 \\
32 \left(x^5+103 x^4+120 x^3+139 x^2+46 x+12\right) & 151 \\
32 \left(x^5+30 x^4+118 x^3+8 x^2+129 x+45\right) & 157 \\
32 \left(x^5+106 x^4+14 x^3+88 x^2+117 x+29\right) & 163 \\
32 \left(x^5+45 x^4+72 x^3+12 x^2+121 x+145\right) & 167 \\
32 \left(x^5+92 x^4+68 x^3+96 x^2+113 x+124\right) & 173 \\
32 \left(x^5+45 x^4+152 x^3+155 x^2+24 x+106\right) & 179 \\
32 \left(x^5+143 x^4+53 x^3+27 x^2+98 x+159\right) & 181 \\
32 \left(x^5+98 x^4+38 x^3+77 x^2+24 x+8\right) & 191 \\
32 \left(x^5+161 x^4+179 x^3+137 x^2+114 x+100\right) & 193 \\
32 (x+18) (x+108) (x+111) (x+136) (x+183) & 197 \\
32 (x+1) (x+38) (x+54) (x+82) (x+120) & 199 \\
32 \left(x^5+120 x^4+97 x^3+55 x^2+101 x+117\right) & 211 \\
32 \left(x^5+195 x^4+85 x^3+192 x^2+117 x+110\right) & 223 \\
32 \left(x^5+137 x^4+98 x^3+25 x^2+222 x+226\right) & 227 \\
32 \left(x^5+132 x^4+168 x^3+177 x^2+8 x+88\right) & 229 \\
32 \left(x^5+170 x^4+85 x^3+15 x^2+172 x+102\right) & 233 \\
32 \left(x^5+108 x^4+72 x^3+143 x^2+6 x+68\right) & 239 \\
32 (x+54) (x+62) (x+84) (x+97) (x+141) & 241 \\
32 \left(x^5+129 x^4+183 x^3+166 x^2+184 x+168\right) & 251 \\
32 \left(x^5+200 x^4+214 x^3+61 x^2+170 x+88\right) & 257 \\
32 (x+138) (x+160) (x+204) (x+206) (x+209) & 263 \\
32 \left(x^5+164 x^4+124 x^3+201 x^2+227 x+122\right) & 269 \\
32 \left(x^5+200 x^4+159 x^3+58 x^2+123 x+241\right) & 271 \\
32 \left(x^5+103 x^4+97 x^3+147 x^2+263 x+8\right) & 277 \\
32 \left(x^5+279 x^4+268 x^2+231 x+153\right) & 281 \\
32 \left(x^5+102 x^4+257 x^3+148 x^2+18 x+75\right) & 283 \\
32 \left(x^5+216 x^4+134 x^3+55 x^2+248 x+25\right) & 293 \\
32 (x+30) (x+35) (x+70) (x+85) (x+224) & 307 \\
32 \left(x^5+19 x^4+292 x^3+117 x^2+92 x+55\right) & 311 \\
32 \left(x^5+285 x^4+27 x^3+129 x^2+209 x+200\right) & 313 \\
32 \left(x^5+219 x^4+265 x^3+180 x^2+79 x+148\right) & 317 \\
32 (x+159) (x+208) (x+245) (x+303) (x+318) & 331 \\
32 \left(x^5+32 x^4+187 x^3+294 x^2+24 x+170\right) & 337 \\
32 \left(x^5+50 x^4+119 x^3+214 x^2+16 x+5\right) & 347 \\
32 \left(x^5+145 x^4+190 x^3+248 x^2+313 x+347\right) & 349 \\
32 (x+60) (x+187) (x+213) (x+253) (x+352) & 353 \\
32 \left(x^5+22 x^4+65 x^3+298 x^2+234 x+108\right) & 359 \\
32 \left(x^5+259 x^4+245 x^3+338 x^2+335 x+28\right) & 367 \\
32 (x+40) (x+54) (x+154) (x+199) (x+343) & 373 \\
32 \left(x^5+256 x^4+302 x^3+70 x^2+113 x+181\right) & 379 \\
32 \left(x^5+303 x^4+105 x^3+190 x^2+129 x+341\right) & 383 \\
32 \left(x^5+224 x^4+377 x^3+46 x^2+37 x+368\right) & 389 \\
32 (x+99) (x+189) (x+289) (x+293) (x+382) & 397 \\
32 \left(x^5+210 x^4+352 x^3+294 x^2+67 x+139\right) & 401 \\
32 \left(x^5+171 x^4+228 x^3+80 x^2+219 x+12\right) & 409 \\
32 (x+52) (x+96) (x+243) (x+261) (x+316) & 419 \\
32 \left(x^5+52 x^4+63 x^3+169 x^2+283 x+275\right) & 421 \\
32 \left(x^5+153 x^4+312 x^3+86 x^2+193 x+375\right) & 431 \\
32 \left(x^5+101 x^4+145 x^3+95 x^2+66 x+108\right) & 433 \\
32 (x+18) (x+97) (x+201) (x+226) (x+305) & 439 \\
32 \left(x^5+344 x^4+121 x^3+359 x^2+142 x+39\right) & 443 \\
32 \left(x^5+278 x^4+141 x^3+358 x^2+441 x+162\right) & 449 \\
32 \left(x^5+246 x^4+76 x^3+71 x^2+304 x+41\right) & 457 \\
32 (x+69) (x+200) (x+261) (x+269) (x+377) & 461 \\
32 (x+36) (x+37) (x+196) (x+224) (x+234) & 463 \\
32 \left(x^5+296 x^4+143 x^3+353 x^2+248 x+333\right) & 467 \\
32 \left(x^5+9 x^4+358 x^3+164 x^2+93 x+433\right) & 479 \\
32 \left(x^5+209 x^4+125 x^3+106 x^2+425 x+456\right) & 487 \\
32 \left(x^5+333 x^4+447 x^3+376 x^2+145 x+387\right) & 491 \\
32 \left(x^5+130 x^4+249 x^3+269 x^2+29 x+402\right) & 499 \\
32 \left(x^5+298 x^4+192 x^3+99 x^2+371 x+422\right) & 503 \\
32 \left(x^5+71 x^4+41 x^3+294 x^2+255 x+3\right) & 509 \\
32 \left(x^5+210 x^4+32 x^3+134 x^2+505 x+464\right) & 521 \\
32 \left(x^5+332 x^4+383 x^3+4 x^2+128 x+220\right) & 523 \\
32 \left(x^5+528 x^4+46 x^3+276 x^2+169 x+503\right) & 541 \\
\end{array}
\right)\]

计算了那么多素数，结果发现只有(1)(1)(1)(1)(1)、(5)这两种情况，很明显这就是个循环群，因此方程
f=32*x^5+3349456*x^4-5941616812296*x^3-585145514845851080*x^2+147013447513276833423286*x+15377302441624829616294559439;
有根式解！

xiaoshuchong

nyy 发表于 2024-1-12 16:39
接上面的代码：

整理一下，可以得到一个清晰的结果：
$$
\begin{eqnarray}
a&=&16557825695384603350011256367\\b&=&7402106235083715173833501130\\c&=&86691265174494495730246967804\\&&90669721303755676965485\\d&=&38769110403114254885491161903\\&&43499942853937801760782\\T_{1,2}&=&\frac{-\left(a-b\sqrt{5}\right)\pm115\sqrt{c-d\sqrt{5}}i}{2}\\T_{3,4}&=&\frac{-\left(a+b\sqrt{5}\right)\pm115\sqrt{c+d\sqrt{5}}i}{2}\\L&=&\left[\begin{array}{cccc}
0 & 0 & 0 & 0\\
1 & 4 & 2 & 3\\
2 & 3 & 4 & 1\\
3 & 2 & 1 & 4\\
4 & 1 & 3 & 2
\end{array}\right],\zeta=\exp\left(\frac{2\pi i}{5}\right)\\x_{j}&=&\frac{-209341+11^{\frac{3}{5}}\sum_{k=1}^{4}\sqrt[5]{T_{k}}\zeta^{L_{jk}}}{10},j=1,\cdots,5
\end{eqnarray}
$$
T1,T2,T3,T4是一个四次方程的解。

nyy

xiaoshuchong 发表于 2024-2-20 09:33
整理一下，可以得到一个清晰的结果：
$$
\begin{eqnarray}

你应该给出供别人验证的代码，来检验你的输出结果

xiaoshuchong

nyy 发表于 2024-4-15 09:35
你应该给出供别人验证的代码，来检验你的输出结果

感谢提醒。
以下是gp/pari代码
test09() = {
    a = 16557825695384603350011256367;
    b = 7402106235083715173833501130;
    c = 22633927789404195625906842766836083950159361724001877;
    d = 10122181481681239360668340460448686303713447708473285;
    t5 = quadunit(20) - 2;

    d2 = (a-b*t5)^2 - 2*11^4*(c-d*t5);
    print(d2);
    d2 = (a+b*t5)^2 - 2*11^4*(c+d*t5);
    print(d2);

    default(realprecision,100);
    c = 8669126517449449573024696780490669721303755676965485;
    d = 3876911040311425488549116190343499942853937801760782;

    T1 = (-a+b*t5+115*sqrt(c-d*t5)*I)/2;
    T2 = (-a+b*t5-115*sqrt(c-d*t5)*I)/2;
    T3 = (-a-b*t5+115*sqrt(c+d*t5)*I)/2;
    T4 = (-a-b*t5-115*sqrt(c+d*t5)*I)/2;
    T = [T1,T2,T3,T4];

    s = prod(n=1,4,x-T[n]);
    print(round(s));
   
    w = exp(2*Pi*I/5);
    W = vector(5,n,w^(n-1));
    Ls = [
        [0,0,0,0],[1,4,2,3],[2,3,4,1],[3,2,1,4],[4,1,3,2]
    ];
    X = List();
    for(n = 1,5,
        x0 = -209341+11^(3/5)*sum(k=1,4,T[k]^0.2*W[Ls[n][k]+1]);
        listput(X,x0/10);
    );
    eqn = prod(n=1,5,x-X[n])*32;
    print(eqn);
    print(round(eqn));
};
test09();

hf2824259

这个五次方程怎么解？

nyy

hf2824259 发表于 2024-6-23 17:05
这个五次方程怎么解？

你写的啥东西，又不能给别人验证！