如何求一个一元八次方程的符号解

chyanog

90# 数学星空
这个可以用三角函数的

数学星空

呵呵!
$1 + 6 x - 12 x^2 - 32 x^3 + 16 x^4 + 32 x^5$=
$32*(x+cos(pi/11))*(x+cos((3*pi)/11))*(x+cos((5*pi)/11))*(x+cos((7*pi)/11))*(x+cos((9*pi)/11))$

数学星空

令$alpha$为方程 $x^10+x^9+x^8+x^7+x^6+x^5+x^4+x^3+x^2+x+1=0$的根
则：$32*x^5+16*x^4-32*x^3-12*x^2+6*x+1$＝
$(-2*x+alpha^2+alpha^9)*(-2*x+alpha^5+alpha^6)*(-2*x+alpha^3+alpha^8)*(2*x+1+alpha^2+alpha^3+alpha^4+alpha^5+alpha^6+alpha^7+alpha^8+alpha^9)*(-2*x+alpha^4+alpha^7)$

数学星空

令\(\alpha\)为方程\(\frac{x^{17}-1}{x-1}=0\)的单位根，则

\(6484591447895521-3109710821335221600x^2+367081320495641587776x^4-16753488126971906739456x^6+385092117683114579108352x^8-5103856857770702945685504x^{10}+42049992724818030982324224x^{12}

-225294153564237402600603648x^{14}+806880282203535092765097984x^{16}-1963620676373502637045186560x^{18}+3275395806763778081820770304x^{20}-3751220300859755209194209280x^{22}+

2928086932006564898870919168x^{24}-1524285912484356749116047360x^{26}+504730181549077650531680256x^{28}-95946081158798277328502784x^{30}+7958661109946400884391936x^{32}=\)

\((1296x^4-1872x^2+648x^2\alpha^6+648x^2\alpha^{11}+262-180\alpha^6+81\alpha^{12}+81\alpha^5-180\alpha^{11})(262-1872x^2+1296x^4-180\alpha^2+81\alpha^4-180\alpha^{15}+81\alpha^{13}+648x^2\alpha^2+648x^2\alpha^{15})(262-1872x^2+1296x^4-180

\alpha^5+81\alpha^{10}-180\alpha^{12}+81\alpha^7+648x^2\alpha^5+648x^2\alpha^{12})

(-181+1872x^2-1296x^4+81\alpha^2+81\alpha^4+81\alpha^3+81\alpha^6+261\alpha^8+81\alpha^5+81\alpha^{10}+81\alpha^{12}+81\alpha^7+81\alpha^{14}+81\alpha^{15}+81\alpha^{13}+81\alpha^{11}+261\alpha^9-648x^2\alpha^8-648\alpha^9x^2)

(-442-261\alpha^2-180\alpha^9-180\alpha^5-180\alpha^6-180\alpha^3-180\alpha^8-180\alpha^4-180\alpha^7+2520x^2-1296x^4-180\alpha^{14}-

261\alpha^{15}+648x^2\alpha^8+648x^2\alpha^5+648x^2\alpha^6+648\alpha^9x^2+648x^2\alpha^7+648x^2\alpha^{10}+648x^2\alpha^{11}+648x^2\alpha^{12}+648x^2\alpha^{13}+648x^2\alpha^{14}+648x^2\alpha^{15}+648x^2\alpha^2+648x^2\alpha^4+648x^2

\alpha^3-180\alpha^{11}-180\alpha^{12}-180\alpha^{13}-180\alpha^{10})(262-1872x^2+1296x^4-180\alpha^4+81\alpha^8-180\alpha^{13}+81\alpha^9+648x^2\alpha^{13}+648x^2\alpha^4)

(262-1872x^2+1296x^4+648x^2\alpha^{10}+648x^2\alpha^7+81\alpha^3-180\alpha^{10}-180\alpha^7+81\alpha^{14})(262-1872x^2+1296x^4-180\alpha^3+81\alpha^6-180\alpha^{14}+81\alpha^{11}+648x^2\alpha^3+648x^2\alpha^{14})\)

xiaoshuchong

数学星空 发表于 2013-1-20 13:57
呵呵，mathe 说到了本质：
[TeX]y^8+12*y^7-72*y^6+24*y^5+65*y^4-12*y^3-18*y^2+1=-(-y^4-6*y^3+7*y^3*sq ...

根号可以开出来，y有如下形式
\[\begin{eqnarray*}
y_{1},y_{2}&=&\frac{-6-3\sqrt{6}-\sqrt{42}-2\sqrt{7}}{4}\pm\frac{2\sqrt{21}+7\sqrt{2}+6\sqrt{3}+3\sqrt{14}}{4}\\y_{3},y_{4}&=&\frac{-6+3\sqrt{6}-\sqrt{42}+2\sqrt{7}}{4}\pm\frac{2\sqrt{21}+7\sqrt{2}-6\sqrt{3}-3\sqrt{14}}{4}\\y_{5},y_{6}&=&\frac{-6-3\sqrt{6}+\sqrt{42}+2\sqrt{7}}{4}\pm\frac{2\sqrt{21}-7\sqrt{2}-6\sqrt{3}+3\sqrt{14}}{4}\\y_{7},y_{8}&=&\frac{-6+3\sqrt{6}+\sqrt{42}-2\sqrt{7}}{4}\pm\frac{2\sqrt{21}-7\sqrt{2}+6\sqrt{3}-3\sqrt{14}}{4}
\end{eqnarray*}\]

nyy

http://magma.maths.usyd.edu.au/calc/

1. R<x> := PolynomialRing(Rationals());
   
2. f := 256*x^8 + 1536*x^7 - 4608*x^6 + 768*x^5 + 1040*x^4 - 96*x^3 - 72*x^2 + 1;
   
3. G, data := GaloisGroup(f);
   
4. TransitiveGroupDescription(G);
   
5. IsSolvable(G);
   
6. print G;
   
7. print data;
   
8. K, roots := SolveByRadicals(f : Name := "r");
   
9. K:Maximal;

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得到

1. E(8)=2[x]2[x]2
   
2. true
   
3. Permutation group G acting on a set of cardinality 8
   
4. Order = 8 = 2^3
   
5.     (1, 2)(3, 8)(4, 5)(6, 7)
   
6.     (1, 3)(2, 8)(4, 7)(5, 6)
   
7.     (1, 4)(2, 5)(3, 7)(6, 8)
   
8. [ -10948731936 + O(167^5), 26672881738 + O(167^5), 34225908539 + O(167^5),
   
9. -24635306430 + O(167^5), 24515533115 + O(167^5), 11958819826 + O(167^5),
   
10. -61661785090 + O(167^5), -127321298 + O(167^5) ]
    
11. 
    
12.   K<r1>
    
13.     |
    
14.     |
    
15.   $1<r2>
    
16.     |
    
17.     |
    
18.   $2<r3>
    
19.     |
    
20.     |
    
21.     Q
    
22. 
    
23. K  : r1^2 + 1/5*(-224*r3 + 1344)*r2 + 114688*r3 - 286720
    
24. $1 : r2^2 + 344064*r3 - 843776
    
25. $2 : r3^2 - 6
    
26. 
    
27. 
    

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根据上面的求解结果。取出

1. K  : r1^2 + 1/5*(-224*r3 + 1344)*r2 + 114688*r3 - 286720
   
2. $1 : r2^2 + 344064*r3 - 843776
   
3. $2 : r3^2 - 6

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对这个联立求解方程组，求解出r1 r2 r3的值，得到
\[\begin{array}{rrr}
\text{r1}\to -64 \sqrt{7 \left(6 \sqrt{2}+5 \sqrt{3}+4 \sqrt{6}+10\right)} & \text{r2}\to -64 \left(7 \sqrt{2}+6 \sqrt{3}\right) & \text{r3}\to -\sqrt{6} \\
\text{r1}\to 64 \sqrt{7 \left(6 \sqrt{2}+5 \sqrt{3}+4 \sqrt{6}+10\right)} & \text{r2}\to -64 \left(7 \sqrt{2}+6 \sqrt{3}\right) & \text{r3}\to -\sqrt{6} \\
\text{r1}\to -64 \sqrt{7 \left(-6 \sqrt{2}-5 \sqrt{3}+4 \sqrt{6}+10\right)} & \text{r2}\to 64 \left(7 \sqrt{2}+6 \sqrt{3}\right) & \text{r3}\to -\sqrt{6} \\
\text{r1}\to 64 \sqrt{7 \left(-6 \sqrt{2}-5 \sqrt{3}+4 \sqrt{6}+10\right)} & \text{r2}\to 64 \left(7 \sqrt{2}+6 \sqrt{3}\right) & \text{r3}\to -\sqrt{6} \\
\text{r1}\to -64 \sqrt{7 \left(-6 \sqrt{2}+5 \sqrt{3}-4 \sqrt{6}+10\right)} & \text{r2}\to 64 \left(7 \sqrt{2}-6 \sqrt{3}\right) & \text{r3}\to \sqrt{6} \\
\text{r1}\to 64 \sqrt{7 \left(-6 \sqrt{2}+5 \sqrt{3}-4 \sqrt{6}+10\right)} & \text{r2}\to 64 \left(7 \sqrt{2}-6 \sqrt{3}\right) & \text{r3}\to \sqrt{6} \\
\text{r1}\to -64 \sqrt{7 \left(6 \sqrt{2}-5 \sqrt{3}-4 \sqrt{6}+10\right)} & \text{r2}\to 64 \left(6 \sqrt{3}-7 \sqrt{2}\right) & \text{r3}\to \sqrt{6} \\
\text{r1}\to 64 \sqrt{7 \left(6 \sqrt{2}-5 \sqrt{3}-4 \sqrt{6}+10\right)} & \text{r2}\to 64 \left(6 \sqrt{3}-7 \sqrt{2}\right) & \text{r3}\to \sqrt{6} \\
\end{array}\]

假设原来的方程的根是如下的根式组合

1. 1/256*r1+1/512*r2+1/8*(3*r3-6)

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然后得到原方程的根如下：
\[\begin{array}{r}
\frac{1}{8} \left(-7 \sqrt{2}-3 \sqrt{3} \left(\sqrt{2}+2\right)-2 \sqrt{7 \left(6 \sqrt{2}+5 \sqrt{3}+4 \sqrt{6}+10\right)}-6\right) \\
\frac{1}{8} \left(-7 \sqrt{2}-3 \sqrt{3} \left(\sqrt{2}+2\right)+2 \sqrt{7 \left(6 \sqrt{2}+5 \sqrt{3}+4 \sqrt{6}+10\right)}-6\right) \\
\frac{1}{8} \left(7 \sqrt{2}+6 \sqrt{3}-3 \sqrt{6}-2 \sqrt{7 \left(-6 \sqrt{2}-5 \sqrt{3}+4 \sqrt{6}+10\right)}-6\right) \\
\frac{1}{8} \left(7 \sqrt{2}+6 \sqrt{3}-3 \sqrt{6}+2 \sqrt{7 \left(-6 \sqrt{2}-5 \sqrt{3}+4 \sqrt{6}+10\right)}-6\right) \\
\frac{1}{8} \left(7 \sqrt{2}-6 \sqrt{3}+3 \sqrt{6}-2 \sqrt{7 \left(-6 \sqrt{2}+5 \sqrt{3}-4 \sqrt{6}+10\right)}-6\right) \\
\frac{1}{8} \left(7 \sqrt{2}-6 \sqrt{3}+3 \sqrt{6}+2 \sqrt{7 \left(-6 \sqrt{2}+5 \sqrt{3}-4 \sqrt{6}+10\right)}-6\right) \\
\frac{1}{8} \left(-7 \sqrt{2}+6 \sqrt{3}+3 \sqrt{6}-2 \sqrt{7 \left(6 \sqrt{2}-5 \sqrt{3}-4 \sqrt{6}+10\right)}-6\right) \\
\frac{1}{8} \left(-7 \sqrt{2}+6 \sqrt{3}+3 \sqrt{6}+2 \sqrt{7 \left(6 \sqrt{2}-5 \sqrt{3}-4 \sqrt{6}+10\right)}-6\right) \\
\end{array}\]

对这8个根求绝对值并且数值化，再排序，得到
{-8.22533, -0.299175, -0.209319, -0.184738, 0.121337, 0.338983, 0.51309, 1.94515}
可以看出8个根都不同，

再检验这8个根是否是方程的根，经过验算，都是

全部mathematica代码如下

1. Clear["Global`*"];(*Clear all variables*)
   
2. (*定义方程*)
   
3. f=256*x^8+1536*x^7-4608*x^6+768*x^5+1040*x^4-96*x^3-72*x^2+1
   
4. $MaxExtraPrecision=2000;
   
5. (*求解根式*)
   
6. ans=Solve[{
   
7.     r1^2+1/5*(-224*r3+1344)*r2+114688*r3-286720,
   
8.     r2^2+344064*r3-843776,
   
9.     r3^2-6
   
10. }==0,{r1,r2,r3}]//FullSimplify;
    
11. ans2=Grid[ans,Alignment->Right](*列表显示*)
    
12. (*假设解是下面的根式组合*)
    
13. sol=1/256*r1+1/512*r2+1/8*(3*r3-6)
    
14. (*把根式代入解*)
    
15. aa=(sol/.ans)//FullSimplify;
    
16. aa2=Grid[Transpose@{aa},Alignment->Right](*列表显示*)
    
17. aaa=Sort@N@aa (*查看根的数值解是否有重复*)
    
18. (*把解代入方程进行数值验证*)
    
19. bb=(f/.x->aa)//N[#,100]&
    
20. cc=Sort@Abs@bb
    

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nyy

nyy 发表于 2024-2-19 13:03
http://magma.maths.usyd.edu.au/calc/

下面来解决\[7 \left(6 \sqrt{2}+5 \sqrt{3}+4 \sqrt{6}+10\right)\]开平方根的问题
假设
\[7 \left(6 \sqrt{2}+5 \sqrt{3}+4 \sqrt{6}+10\right)-\left(a+\sqrt{2} b+\sqrt{3} c+\sqrt{6} d\right)^2==0\]
展开，得到系数，并且令系数等于零，得到
{70 - a^2 - 2 b^2 - 3 c^2 - 6 d^2, 35 - 2 a c - 4 b d, 42 - 2 a b - 6 c d, 28 - 2 b c - 2 a d}=0
解这个方程组，然后只保留让平方根大于零的解
得到
\[\begin{array}{rrrr}
a\to 3 \sqrt{\frac{7}{2}} & b\to \sqrt{\frac{7}{2}} & c\to \sqrt{\frac{7}{2}} & d\to \sqrt{\frac{7}{2}} \\
a\to \sqrt{7} & b\to \frac{3 \sqrt{7}}{2} & c\to \sqrt{7} & d\to \frac{\sqrt{7}}{2} \\
a\to \sqrt{\frac{21}{2}} & b\to \sqrt{\frac{21}{2}} & c\to \sqrt{\frac{21}{2}} & d\to \sqrt{\frac{7}{6}} \\
a\to \sqrt{21} & b\to \frac{\sqrt{21}}{2} & c\to \sqrt{\frac{7}{3}} & d\to \frac{\sqrt{21}}{2} \\
a\to \frac{1}{2} \sqrt{70-7 \sqrt{3 \left(9-4 \sqrt{2}\right)}} & b\to \frac{1}{2} \sqrt{\frac{7}{2} \left(\sqrt{3 \left(9-4 \sqrt{2}\right)}+10\right)} & c\to \frac{1}{2} \sqrt{\frac{7}{3} \left(\sqrt{3 \left(4 \sqrt{2}+9\right)}+10\right)} & d\to \frac{1}{12} \sqrt{420-42 \left(\sqrt{3}+2 \sqrt{6}\right)} \\
a\to \frac{1}{2} \sqrt{7 \left(\sqrt{3 \left(9-4 \sqrt{2}\right)}+10\right)} & b\to \frac{1}{2} \sqrt{35-\frac{7}{2} \sqrt{3 \left(9-4 \sqrt{2}\right)}} & c\to \frac{1}{6} \sqrt{210-21 \left(\sqrt{3}+2 \sqrt{6}\right)} & d\to \frac{1}{2} \sqrt{\frac{7}{6} \left(\sqrt{3 \left(4 \sqrt{2}+9\right)}+10\right)} \\
a\to \frac{1}{2} \sqrt{70-7 \sqrt{3 \left(4 \sqrt{2}+9\right)}} & b\to \frac{1}{2} \sqrt{\frac{7}{2} \left(\sqrt{3 \left(4 \sqrt{2}+9\right)}+10\right)} & c\to \frac{1}{2} \sqrt{\frac{7}{3} \left(\sqrt{3 \left(9-4 \sqrt{2}\right)}+10\right)} & d\to \frac{1}{2} \sqrt{\frac{7}{6} \left(\sqrt{3}-2 \sqrt{6}+10\right)} \\
a\to \frac{1}{2} \sqrt{7 \left(\sqrt{3 \left(4 \sqrt{2}+9\right)}+10\right)} & b\to \frac{1}{2} \sqrt{35-\frac{7}{2} \sqrt{3 \left(4 \sqrt{2}+9\right)}} & c\to \frac{1}{2} \sqrt{\frac{7}{3} \left(\sqrt{3}-2 \sqrt{6}+10\right)} & d\to \frac{1}{2} \sqrt{\frac{7}{6} \left(\sqrt{3 \left(9-4 \sqrt{2}\right)}+10\right)} \\
\end{array}\]

把上面的解代入\[7 \left(6 \sqrt{2}+5 \sqrt{3}+4 \sqrt{6}+10\right)-\left(a+\sqrt{2} b+\sqrt{3} c+\sqrt{6} d\right)^2==0\]，都等于零，证明只是不同的表达形式。
选择第一组就可以了！
具体的最后结果\(\sqrt{7}+\sqrt{21}+3 \sqrt{\frac{7}{2}}+\sqrt{\frac{21}{2}}\)(也就是\(7 \left(6 \sqrt{2}+5 \sqrt{3}+4 \sqrt{6}+10\right)\)的算术平方根)
余下同理！





补充内容 (2024-2-20 09:14):
Clear["Global`*"];(*Clear all variables*)
(*假设表达式如下*)
root=(a+b*Sqrt[2]+c*Sqrt[3]+d*Sqrt[6])
f=7*(10+6*Sqrt[2]+5*Sqrt[3]+4*Sqrt[6])-root^2//Expand
(*求解出系数*)
aaa=Flatten@Coef...

nyy

1. Clear["Global`*"];(*Clear all variables*)
   
2. (*定义方程*)
   
3. f=256*x^8+1536*x^7-4608*x^6+768*x^5+1040*x^4-96*x^3-72*x^2+1
   
4. (*对着答案凑结果*)
   
5. Factor[f,Extension->{Sqrt[2],Sqrt[3],Sqrt[6],Sqrt[7],Sqrt[14],Sqrt[42]}]
   

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得到
\[
\frac{
\left(-8 x+7 \sqrt{2}+6 \sqrt{3}-3 \sqrt{6}+2 \sqrt{7}-3 \sqrt{14}-2 \sqrt{21}+\sqrt{42}-6\right) \\
\left(-8 x+7 \sqrt{2}-6 \sqrt{3}+3 \sqrt{6}-2 \sqrt{7}+3 \sqrt{14}-2 \sqrt{21}+\sqrt{42}-6\right) \\
\left(-8 x-7 \sqrt{2}+6 \sqrt{3}+3 \sqrt{6}-2 \sqrt{7}-3 \sqrt{14}+2 \sqrt{21}+\sqrt{42}-6\right) \\
\left(-8 x-7 \sqrt{2}-6 \sqrt{3}-3 \sqrt{6}+2 \sqrt{7}+3 \sqrt{14}+2 \sqrt{21}+\sqrt{42}-6\right) \\
\left(8 x-7 \sqrt{2}-6 \sqrt{3}+3 \sqrt{6}+2 \sqrt{7}-3 \sqrt{14}-2 \sqrt{21}+\sqrt{42}+6\right)\\
\left(8 x-7 \sqrt{2}+6 \sqrt{3}-3 \sqrt{6}-2 \sqrt{7}+3 \sqrt{14}-2 \sqrt{21}+\sqrt{42}+6\right) \\
\left(8 x+7 \sqrt{2}-6 \sqrt{3}-3 \sqrt{6}-2 \sqrt{7}-3 \sqrt{14}+2 \sqrt{21}+\sqrt{42}+6\right) \\
\left(8 x+7 \sqrt{2}+6 \sqrt{3}+3 \sqrt{6}+2 \sqrt{7}+3 \sqrt{14}+2 \sqrt{21}+\sqrt{42}+6\right)}{65536}
\]

nyy

mathe 发表于 2013-1-21 15:58
Gap的radiroot扩展
http://www.gap-system.org/Manuals/pkg/radiroot/htm/CHAP002.htm

你居然也知道这个，而且比我早了十年这样

nyy

mathe 发表于 2013-1-20 14:13
我是借助计算机来分析可解群。为此，我们可以因子分解f(x)(mod p),其中p是素数。我发现对不同的素数，结果 ...

如果我没猜错，你应该按照这猜结果的

mathematica代码

1. Clear["Global`*"];(*Clear all variables*)
   
2. f=256*x^8 + 1536*x^7 - 4608*x^6 + 768*x^5 + 1040*x^4 - 96*x^3 - 72*x^2 + 1
   
3. aa={Factor[f,Modulus->#],#}&/@Prime@Range[20,100];
   
4. MatrixForm[aa]
   

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求解结果
\[\left(
\begin{array}{cc}
43 \left(x^2+13 x+13\right) \left(x^2+19 x+14\right) \left(x^2+48 x+9\right) \left(x^2+68 x+36\right) & 71 \\
37 \left(x^2+9 x+65\right) \left(x^2+33 x+69\right) \left(x^2+53 x+67\right) \left(x^2+57 x+19\right) & 73 \\
19 \left(x^2+23 x+32\right) \left(x^2+25 x+3\right) \left(x^2+49 x+28\right) \left(x^2+67 x+52\right) & 79 \\
7 \left(x^2+50 x+47\right) \left(x^2+52 x+22\right) \left(x^2+75 x+80\right) \left(x^2+78 x+13\right) & 83 \\
78 \left(x^2+x+16\right) \left(x^2+14 x+43\right) \left(x^2+35 x+63\right) \left(x^2+45 x+8\right) & 89 \\
62 \left(x^2+3 x+34\right) \left(x^2+49 x+74\right) \left(x^2+67 x+10\right) \left(x^2+81 x+77\right) & 97 \\
54 \left(x^2+23 x+100\right) \left(x^2+33 x+19\right) \left(x^2+63 x+73\right) \left(x^2+89 x+62\right) & 101 \\
50 \left(x^2+17 x+50\right) \left(x^2+59 x+30\right) \left(x^2+64 x+53\right) \left(x^2+72 x+73\right) & 103 \\
42 \left(x^2+6 x+18\right) \left(x^2+51 x+46\right) \left(x^2+67 x+13\right) \left(x^2+96 x+33\right) & 107 \\
38 \left(x^2+4 x+17\right) \left(x^2+14 x+106\right) \left(x^2+27 x+76\right) \left(x^2+70 x+15\right) & 109 \\
30 \left(x^2+3 x+17\right) \left(x^2+9 x+86\right) \left(x^2+32 x+96\right) \left(x^2+75 x+27\right) & 113 \\
2 \left(x^2+6 x+103\right) \left(x^2+20 x+80\right) \left(x^2+53 x+42\right) \left(x^2+54 x+89\right) & 127 \\
125 \left(x^2+33 x+107\right) \left(x^2+118 x+4\right) \left(x^2+121 x+7\right) \left(x^2+127 x+9\right) & 131 \\
119 \left(x^2+49 x+85\right) \left(x^2+51 x+41\right) \left(x^2+79 x+52\right) \left(x^2+101 x+96\right) & 137 \\
117 \left(x^2+23 x+39\right) \left(x^2+71 x+73\right) \left(x^2+90 x+40\right) \left(x^2+100 x+110\right) & 139 \\
107 \left(x^2+23 x+1\right) \left(x^2+25 x+117\right) \left(x^2+34 x+5\right) \left(x^2+73 x+10\right) & 149 \\
105 \left(x^2+45 x+140\right) \left(x^2+80 x+137\right) \left(x^2+84 x+91\right) \left(x^2+99 x+6\right) & 151 \\
99 \left(x^2+87 x+102\right) \left(x^2+124 x+82\right) \left(x^2+132 x+14\right) \left(x^2+134 x+119\right) & 157 \\
93 \left(x^2+101 x+40\right) \left(x^2+127 x+39\right) \left(x^2+129 x+99\right) \left(x^2+138 x+68\right) & 163 \\
89 (x+45) (x+47) (x+59) (x+61) (x+75) (x+118) (x+130) (x+139) & 167 \\
83 \left(x^2+19 x+49\right) \left(x^2+43 x+53\right) \left(x^2+123 x+47\right) \left(x^2+167 x+112\right) & 173 \\
77 \left(x^2+62 x+102\right) \left(x^2+63 x+69\right) \left(x^2+109 x+17\right) \left(x^2+130 x+173\right) & 179 \\
75 \left(x^2+2 x+11\right) \left(x^2+23 x+145\right) \left(x^2+62 x+21\right) \left(x^2+100 x+7\right) & 181 \\
65 \left(x^2+10 x+172\right) \left(x^2+42 x+155\right) \left(x^2+65 x+127\right) \left(x^2+80 x+120\right) & 191 \\
63 (x+8) (x+26) (x+51) (x+53) (x+81) (x+95) (x+120) (x+151) & 193 \\
59 \left(x^2+2 x+111\right) \left(x^2+107 x+154\right) \left(x^2+123 x+167\right) \left(x^2+168 x+143\right) & 197 \\
57 \left(x^2+34 x+188\right) \left(x^2+81 x+11\right) \left(x^2+98 x+26\right) \left(x^2+191 x+173\right) & 199 \\
45 \left(x^2+116 x+93\right) \left(x^2+167 x+183\right) \left(x^2+175 x+138\right) \left(x^2+181 x+115\right) & 211 \\
33 \left(x^2+x+71\right) \left(x^2+86 x+138\right) \left(x^2+166 x+85\right) \left(x^2+199 x+152\right) & 223 \\
29 \left(x^2+41 x+132\right) \left(x^2+112 x+144\right) \left(x^2+152 x+64\right) \left(x^2+155 x+110\right) & 227 \\
27 \left(x^2+24 x+174\right) \left(x^2+80 x+107\right) \left(x^2+136 x+159\right) \left(x^2+224 x+21\right) & 229 \\
23 \left(x^2+27 x+164\right) \left(x^2+28 x+191\right) \left(x^2+46 x+69\right) \left(x^2+138 x+42\right) & 233 \\
17 \left(x^2+71 x+143\right) \left(x^2+92 x+163\right) \left(x^2+138 x+91\right) \left(x^2+183 x+82\right) & 239 \\
15 \left(x^2+80 x+132\right) \left(x^2+84 x+56\right) \left(x^2+87 x+93\right) \left(x^2+237 x+202\right) & 241 \\
5 \left(x^2+82 x+80\right) \left(x^2+101 x+79\right) \left(x^2+130 x+75\right) \left(x^2+195 x+13\right) & 251 \\
256 \left(x^2+72 x+205\right) \left(x^2+78 x+142\right) \left(x^2+141 x+198\right) \left(x^2+229 x+94\right) & 257 \\
256 \left(x^2+22 x+232\right) \left(x^2+50 x+43\right) \left(x^2+94 x+138\right) \left(x^2+103 x+114\right) & 263 \\
256 \left(x^2+21 x+203\right) \left(x^2+156 x+163\right) \left(x^2+173 x+121\right) \left(x^2+194 x+187\right) & 269 \\
256 \left(x^2+69 x+61\right) \left(x^2+84 x+210\right) \left(x^2+125 x+133\right) \left(x^2+270 x+138\right) & 271 \\
256 \left(x^2+9 x+213\right) \left(x^2+44 x+163\right) \left(x^2+107 x+38\right) \left(x^2+123 x+136\right) & 277 \\
256 \left(x^2+59 x+88\right) \left(x^2+123 x+193\right) \left(x^2+125 x+94\right) \left(x^2+261 x+187\right) & 281 \\
256 \left(x^2+172 x+195\right) \left(x^2+177 x+174\right) \left(x^2+225 x+227\right) \left(x^2+281 x+237\right) & 283 \\
256 \left(x^2+70 x+245\right) \left(x^2+256 x+276\right) \left(x^2+270 x+227\right) \left(x^2+289 x+279\right) & 293 \\
256 \left(x^2+19 x+246\right) \left(x^2+147 x+58\right) \left(x^2+206 x+112\right) \left(x^2+248 x+182\right) & 307 \\
256 (x+47) (x+101) (x+169) (x+228) (x+231) (x+233) (x+257) (x+295) & 311 \\
256 \left(x^2+129 x+188\right) \left(x^2+251 x+84\right) \left(x^2+271 x+296\right) \left(x^2+294 x+59\right) & 313 \\
256 \left(x^2+130 x+172\right) \left(x^2+135 x+309\right) \left(x^2+172 x+191\right) \left(x^2+203 x+263\right) & 317 \\
256 \left(x^2+38 x+233\right) \left(x^2+106 x+134\right) \left(x^2+257 x+175\right) \left(x^2+267 x+287\right) & 331 \\
256 (x+28) (x+40) (x+70) (x+85) (x+87) (x+108) (x+278) (x+321) & 337 \\
256 \left(x^2+89 x+79\right) \left(x^2+116 x+14\right) \left(x^2+199 x+317\right) \left(x^2+296 x+287\right) & 347 \\
256 \left(x^2+48 x+273\right) \left(x^2+60 x+95\right) \left(x^2+101 x+160\right) \left(x^2+146 x+173\right) & 349 \\
256 \left(x^2+99 x+191\right) \left(x^2+102 x+199\right) \left(x^2+199 x+101\right) \left(x^2+312 x+35\right) & 353 \\
256 \left(x^2+44 x+275\right) \left(x^2+72 x+160\right) \left(x^2+89 x+166\right) \left(x^2+160 x+118\right) & 359 \\
256 \left(x^2+12 x+332\right) \left(x^2+38 x+35\right) \left(x^2+84 x+251\right) \left(x^2+239 x+116\right) & 367 \\
256 \left(x^2+100 x+20\right) \left(x^2+138 x+47\right) \left(x^2+218 x+208\right) \left(x^2+296 x+94\right) & 373 \\
256 \left(x^2+4 x+49\right) \left(x^2+19 x+241\right) \left(x^2+30 x+178\right) \left(x^2+332 x+102\right) & 379 \\
256 (x+34) (x+48) (x+117) (x+153) (x+232) (x+292) (x+300) (x+362) & 383 \\
256 \left(x^2+95 x+91\right) \left(x^2+173 x+211\right) \left(x^2+177 x+261\right) \left(x^2+339 x+199\right) & 389 \\
256 \left(x^2+88 x+161\right) \left(x^2+344 x+309\right) \left(x^2+372 x+49\right) \left(x^2+393 x+278\right) & 397 \\
256 \left(x^2+37 x+359\right) \left(x^2+107 x+219\right) \left(x^2+282 x+42\right) \left(x^2+382 x+182\right) & 401 \\
256 \left(x^2+43 x+374\right) \left(x^2+234 x+65\right) \left(x^2+273 x+224\right) \left(x^2+274 x+156\right) & 409 \\
256 \left(x^2+11 x+274\right) \left(x^2+38 x+120\right) \left(x^2+79 x+2\right) \left(x^2+297 x+19\right) & 419 \\
256 \left(x^2+48 x+21\right) \left(x^2+88 x+193\right) \left(x^2+114 x+108\right) \left(x^2+177 x+95\right) & 421 \\
256 \left(x^2+38 x+278\right) \left(x^2+73 x+387\right) \left(x^2+157 x+401\right) \left(x^2+169 x+228\right) & 431 \\
256 \left(x^2+65 x+138\right) \left(x^2+83 x+141\right) \left(x^2+91 x+321\right) \left(x^2+200 x+267\right) & 433 \\
256 \left(x^2+286 x+343\right) \left(x^2+302 x+2\right) \left(x^2+305 x+437\right) \left(x^2+430 x+96\right) & 439 \\
256 \left(x^2+119 x+374\right) \left(x^2+234 x+24\right) \left(x^2+257 x+428\right) \left(x^2+282 x+63\right) & 443 \\
256 \left(x^2+21 x+50\right) \left(x^2+142 x+399\right) \left(x^2+326 x+197\right) \left(x^2+415 x+252\right) & 449 \\
256 (x+78) (x+96) (x+156) (x+238) (x+280) (x+281) (x+286) (x+419) & 457 \\
256 \left(x^2+126 x+71\right) \left(x^2+152 x+229\right) \left(x^2+236 x+28\right) \left(x^2+414 x+365\right) & 461 \\
256 \left(x^2+173 x+330\right) \left(x^2+211 x+288\right) \left(x^2+271 x+21\right) \left(x^2+277 x+52\right) & 463 \\
256 \left(x^2+6 x+186\right) \left(x^2+28 x+305\right) \left(x^2+50 x+356\right) \left(x^2+389 x+83\right) & 467 \\
256 (x+109) (x+118) (x+234) (x+251) (x+274) (x+280) (x+294) (x+362) & 479 \\
256 \left(x^2+81 x+227\right) \left(x^2+177 x+9\right) \left(x^2+238 x+301\right) \left(x^2+484 x+190\right) & 487 \\
256 \left(x^2+101 x+20\right) \left(x^2+237 x+201\right) \left(x^2+241 x+38\right) \left(x^2+409 x+235\right) & 491 \\
256 \left(x^2+63 x+40\right) \left(x^2+94 x+28\right) \left(x^2+146 x+137\right) \left(x^2+202 x+46\right) & 499 \\
256 (x+4) (x+27) (x+99) (x+189) (x+214) (x+292) (x+316) (x+374) & 503 \\
256 \left(x^2+185 x+154\right) \left(x^2+226 x+76\right) \left(x^2+238 x+44\right) \left(x^2+375 x+491\right) & 509 \\
256 \left(x^2+36 x+157\right) \left(x^2+280 x+284\right) \left(x^2+334 x+309\right) \left(x^2+398 x+28\right) & 521 \\
256 \left(x^2+290 x+149\right) \left(x^2+377 x+466\right) \left(x^2+445 x+105\right) \left(x^2+463 x+310\right) & 523 \\
256 \left(x^2+74 x+222\right) \left(x^2+212 x+446\right) \left(x^2+299 x+253\right) \left(x^2+503 x+157\right) & 541 \\
\end{array}
\right)\]

你看到这个结果后，看到只有单位元与2*2*2*2，然后猜测群的结构是Z2*Z2*Z2*Z2，不过猜测的结果是错误的.

证据
http://magma.maths.usyd.edu.au/calc/
上代码

1. R<x> := PolynomialRing(Rationals());
   
2. f := 256*x^8 + 1536*x^7 - 4608*x^6 + 768*x^5 + 1040*x^4 - 96*x^3 - 72*x^2 + 1;
   
3. G, data := GaloisGroup(f);
   
4. TransitiveGroupDescription(G);
   
5. IsSolvable(G);
   
6. print G;
   
7. print data;

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输出结果

1. E(8)=2[x]2[x]2
   
2. true
   
3. Permutation group G acting on a set of cardinality 8
   
4. Order = 8 = 2^3 //这个是群的阶，一共八个元素
   
5. //下面这三个，应该是群的生成元（只是我猜测）
   
6.     (1, 2)(3, 8)(4, 5)(6, 7)
   
7.     (1, 3)(2, 8)(4, 7)(5, 6)
   
8.     (1, 4)(2, 5)(3, 7)(6, 8)
   
9. [ -10948731936 + O(167^5), 26672881738 + O(167^5), 34225908539 + O(167^5),
   
10. -24635306430 + O(167^5), 24515533115 + O(167^5), 11958819826 + O(167^5),
    
11. -61661785090 + O(167^5), -127321298 + O(167^5) ]
    

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群里面的所有元素如下：

1. [
   
2.     Id(G),  //这个是单位元的意思，恒等变换
   
3.     (1, 2)(3, 8)(4, 5)(6, 7),
   
4.     (1, 3)(2, 8)(4, 7)(5, 6),
   
5.     (1, 4)(2, 5)(3, 7)(6, 8),
   
6.     (1, 8)(2, 3)(4, 6)(5, 7),
   
7.     (1, 5)(2, 4)(3, 6)(7, 8),
   
8.     (1, 7)(2, 6)(3, 4)(5, 8),
   
9.     (1, 6)(2, 7)(3, 5)(4, 8)
   
10. ]

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