science123 发表于 2014-11-25 19:00
53楼自己把y解出来看看？
(((sqrt(27*y^2+500)/(2*3^(3/2))+y/2)^(1/3)-5/(3*(sqrt(27*y^2+500)/(2*3^ ...
我听不懂你的逻辑。
消去x后，得到的仍然是 一元五次方程，你怎么就不负责任的直接给出结果了呢？
更让我不解的是，你却反过来问我怎么解。
你要知道 你变换的目的是通过“降次”的手段 解五次方程，你这样把问题转移到一个新的五次方程，其意义何在？
$y^5-100y^3-280y^2-2100y-4008=0$
science123 发表于 2014-11-25 19:29
最后的解还是x^5-5*x-2=0的根，而这个是伽罗瓦解不出来的。
伽罗瓦解不出来，哪你的方法解出没有？
y（大概其等于-2.06195），是那个方程的根呀？
本帖最后由 mathematica 于 2021-5-14 14:23 编辑
//变量是t
P<t>:=PolynomialAlgebra(Rationals());
f:=1*t^5+10*t^2-15*t+6;
f:=f*(t^2+3*t+2);
//注意多项式变量是t
f:=32*t^5+3349456*t^4-5941616812296*t^3-585145514845851080*t^2+147013447513276833423286*t+15377302441624829616294559439;
G:=GaloisGroup(f);
print G;
//看群是否可解
IsSolvable(G);
输出结果：
Permutation group G acting on a set of cardinality 5
Order = 5
(1, 4, 2, 5, 3)
http://magma.maths.usyd.edu.au/calc/
mathematica 发表于 2021-5-14 11:21
输出结果：
Permutation group G acting on a set of cardinality 5
P<x>:=PolynomialRing(IntegerRing());
//下面多项式有根式解
f:=32*x^5+3349456*x^4-5941616812296*x^3-585145514845851080*x^2+147013447513276833423286*x+15377302441624829616294559439;
K,R:=SolveByRadicals(f:Name:="K.");
K:Maximal;
输出结果：
K<K.1>
|
|
$1<K.2>
|
|
$2<K.3>
|
|
Q
K: K.1^5 + (-125444641510450806399613511240171520*K.3 +
281293318510921822495181603202744320)*K.2 -
5165392015600608816069287610432880640*K.3 +
11554503262499979688078203800506597376
$1 : K.2^2 + 128*K.3 + 640
$2 : K.3^2 - 5
本帖最后由 xiaoshuchong 于 2022-3-21 15:27 编辑
令$y=2x$, 则y满足
\[\begin{eqnarray*}
0&=&y^{5}+209341y^{4}-742702101537y^{3}-146286378711462770y^{2}+\\&&73506723756638416711643y+15377302441624829616294559439,(1)
\end{eqnarray*}\]
若$\xi=2\cos\frac{2\pi}{11}$, 则y可表示为$\xi$的四次多项式，即
\
将公式(2)代入公式(1), 可以得到如下五次方程
\
此即$\xi$的最小多项式。
另外，y满足的五次方程的预解式为如下四次方程
\[\begin{eqnarray*}
0&=&R^{4}+\sum_{n=1}^{4}a_{n}R^{4-n}\\a_{1}&=&44076932001113814117729964448954\\a_{2}&=&58743022005372134919929822599143\\&&1012397390749332269918806695106\\a_{3}&=&4878675382757908137915342103443\\&&1871297562268224701487968238890\\&&02996397938213898703855493979\\a_{4}&=&3191249325094399115300187772128\\&&0691898716112816319796030395651\\&&1328374816632053028889864760490\\&&375291885473397252540625951\\&=&795778400695851902616991^{5}
\end{eqnarray*}\]
让我来求解一下这个问题！
在下面网址输入
http://magma.maths.usyd.edu.au/calc/
R<x> := PolynomialRing(Rationals());
f:=32*x^5+3349456*x^4-5941616812296*x^3-585145514845851080*x^2+147013447513276833423286*x+15377302441624829616294559439;
G, data := GaloisGroup(f);
TransitiveGroupDescription(G);
IsSolvable(G);
print G;
K:= SolveByRadicals(f : Name := "r");
K:Maximal;
得到输出结果
C(5) = 5
true
Permutation group G acting on a set of cardinality 5
Order = 5
(1, 4, 2, 5, 3)
K<r1>
|
|
$1<r2>
|
|
$2<r3>
|
|
Q
K: r1^5 + (-125444641510450806399613511240171520*r3 +
281293318510921822495181603202744320)*r2 -
5165392015600608816069287610432880640*r3 +
11554503262499979688078203800506597376
$1 : r2^2 + 128*r3 + 640
$2 : r3^2 - 5
取出其中的
K: r1^5 + (-125444641510450806399613511240171520*r3 +
281293318510921822495181603202744320)*r2 -
5165392015600608816069287610432880640*r3 +
11554503262499979688078203800506597376
$1 : r2^2 + 128*r3 + 640
$2 : r3^2 - 5
联立方程组求解，然后验根。
输入mathematica代码
Clear["Global`*"];(*Clear all variables*)
(*定义方程*)
f=32*x^5+3349456*x^4-5941616812296*x^3-585145514845851080*x^2+147013447513276833423286*x+15377302441624829616294559439
$MaxExtraPrecision =2000;
(*求解根式*)
ans=Solve[{
r1^5+(-125444641510450806399613511240171520*r3+281293318510921822495181603202744320)*r2-5165392015600608816069287610432880640*r3+11554503262499979688078203800506597376,
r2^2+128*r3+640,
r3^2-5
}==0,{r1,r2,r3}]//Simplify;
(*假设解是下面的根式组合*)
sol=(1/5895341351523439674917634854012783589205785342439461517799030059009688232603963881217327104*(613779816943386520735179499432241945095623257886183552204845067579*r3+1401833861594827316193960697258263752005273636699602569940809290618)*r2+1/3684588344702149796823521783757989743253615839024663448624393786881055145377477425760829440*(-1957422353603082441495616556093524740640525031285670498653567856941*r3-5132125680422280685580804916372624186156802396080744540933143565919))*r1^4+(1/280125214079138613865194819283505534154445180150084665851852070584320*(900957634111167014122646048052069894243304497897479*r3+2038847774373034163069683334318249571474043695661000)*r2+1/35015651759892326733149352410438191769305647518760583231481508823040*(-76890511170265658053647893877023837806007692410317*r3-154733658423379615408545465654361507773253477693091))*r1^3+(1/29283172584015339048737829047789333720540856320*(94118720118901230891085096317849939*r3+289847623426031299198396552731049130)*r2+1/3660396573001917381092228630973666715067607040*(2680905286401673045006106350407138389*r3+6020215320576886946232982804028881471))*r1^2+1/160*r1-209341/10;
(*把根式代入解*)
aa=(sol/.ans);
(*把解代入方程进行数值验证*)
bb=(f/.x->aa)//N[#,100]&
cc=Sort@Abs@bb
验根结果误差非常小，“证明”就是要求的方程的根
{0.*10^-2071, 0.*10^-2071, 0.*10^-2071, 0.*10^-2071, 0.*10^-2071,
0.*10^-2071, 0.*10^-2071, 0.*10^-2071, 0.*10^-2071, 0.*10^-2071,
0.*10^-2071, 0.*10^-2070, 0.*10^-2070, 0.*10^-2070, 0.*10^-2070,
0.*10^-2070, 0.*10^-2070, 0.*10^-2070, 0.*10^-2070, 0.*10^-2069}
理论上应该有五个根，但是去重应该困难（我就没去）。列举出其中的一个根
-(209341/10) -
1/10 (-11)^(
3/5) (1/2 (-16557825695384603350011256367 +
7402106235083715173833501130 Sqrt -
115 I Sqrt[
8669126517449449573024696780490669721303755676965485 -
3876911040311425488549116190343499942853937801760782 Sqrt[
5]]))^(1/5) -
1408 (-11)^(1/5) 2^(
3/5) (-16557825695384603350011256367 +
7402106235083715173833501130 Sqrt -
115 I Sqrt[
8669126517449449573024696780490669721303755676965485 -
3876911040311425488549116190343499942853937801760782 Sqrt])^(
2/5) ((I Sqrt[
8669126517449449573024696780490669721303755676965485 -
3876911040311425488549116190343499942853937801760782 Sqrt[
5]] (112166606653350738362971117 +
50021450333620363744415187 Sqrt[
5]) (289847623426031299198396552731049130 +
94118720118901230891085096317849939 Sqrt))/
64624466885202183079915011863734625021224068090281978641398456898\
589021484851301126560268605440 + (
6020215320576886946232982804028881471 +
2680905286401673045006106350407138389 Sqrt)/
3660396573001917381092228630973666715067607040) +
22528 (-11)^(4/5) 2^(
2/5) (-16557825695384603350011256367 +
7402106235083715173833501130 Sqrt -
115 I Sqrt[
8669126517449449573024696780490669721303755676965485 -
3876911040311425488549116190343499942853937801760782 Sqrt])^(
3/5) ((-154733658423379615408545465654361507773253477693091 -
76890511170265658053647893877023837806007692410317 Sqrt)/
350156517598923267331493524104381917693056475187605832314815088230\
40 + (I Sqrt[
8669126517449449573024696780490669721303755676965485 -
3876911040311425488549116190343499942853937801760782 Sqrt[
5]] (112166606653350738362971117 +
50021450333620363744415187 Sqrt[
5]) (2038847774373034163069683334318249571474043695661000 +
900957634111167014122646048052069894243304497897479 Sqrt[
5]))/6182029139441410688567213761151929925575573454601796746\
78966423347976489634426562708126931155027729600046379973181440) +
3964928 (-11)^(2/5) 2^(
1/5) (-16557825695384603350011256367 +
7402106235083715173833501130 Sqrt -
115 I Sqrt[
8669126517449449573024696780490669721303755676965485 -
3876911040311425488549116190343499942853937801760782 Sqrt])^(
4/5) ((-513212568042228068558080491637262418615680239608074454093314\
3565919 -
195742235360308244149561655609352474064052503128567049865356785\
6941 Sqrt)/
36845883447021497968235217837579897432536158390246634486243937868\
81055145377477425760829440 + (I Sqrt[
8669126517449449573024696780490669721303755676965485 -
3876911040311425488549116190343499942853937801760782 Sqrt[
5]] (112166606653350738362971117 +
50021450333620363744415187 Sqrt[
5]) (1401833861594827316193960697258263752005273636699602569\
940809290618 +
6137798169433865207351794994322419450956232578861835522048450\
67579 Sqrt))/
13010314741526848079880729508186937456593895823272362531230819703\
6825683808152694389596902327871733916100917438956973471500490589390299\
46368)
LaTeX输出结果如下：
\[-\frac{1}{10} (-11)^{3/5} \sqrt{\frac{1}{2} \left(7402106235083715173833501130 \sqrt{5}-115 i \sqrt{8669126517449449573024696780490669721303755676965485-3876911040311425488549116190343499942853937801760782 \sqrt{5}}-16557825695384603350011256367\right)}+22528 (-11)^{4/5} 2^{2/5} \left(\frac{-76890511170265658053647893877023837806007692410317 \sqrt{5}-154733658423379615408545465654361507773253477693091}{35015651759892326733149352410438191769305647518760583231481508823040}+\frac{i \sqrt{8669126517449449573024696780490669721303755676965485-3876911040311425488549116190343499942853937801760782 \sqrt{5}} \left(50021450333620363744415187 \sqrt{5}+112166606653350738362971117\right) \left(900957634111167014122646048052069894243304497897479 \sqrt{5}+2038847774373034163069683334318249571474043695661000\right)}{618202913944141068856721376115192992557557345460179674678966423347976489634426562708126931155027729600046379973181440}\right) \left(7402106235083715173833501130 \sqrt{5}-115 i \sqrt{8669126517449449573024696780490669721303755676965485-3876911040311425488549116190343499942853937801760782 \sqrt{5}}-16557825695384603350011256367\right)^{3/5}+3964928 (-11)^{2/5} \sqrt{2} \left(\frac{-1957422353603082441495616556093524740640525031285670498653567856941 \sqrt{5}-5132125680422280685580804916372624186156802396080744540933143565919}{3684588344702149796823521783757989743253615839024663448624393786881055145377477425760829440}+\frac{i \sqrt{8669126517449449573024696780490669721303755676965485-3876911040311425488549116190343499942853937801760782 \sqrt{5}} \left(50021450333620363744415187 \sqrt{5}+112166606653350738362971117\right) \left(613779816943386520735179499432241945095623257886183552204845067579 \sqrt{5}+1401833861594827316193960697258263752005273636699602569940809290618\right)}{13010314741526848079880729508186937456593895823272362531230819703682568380815269438959690232787173391610091743895697347150049058939029946368}\right) \left(7402106235083715173833501130 \sqrt{5}-115 i \sqrt{8669126517449449573024696780490669721303755676965485-3876911040311425488549116190343499942853937801760782 \sqrt{5}}-16557825695384603350011256367\right)^{4/5}-1408 \sqrt{-11} 2^{3/5} \left(7402106235083715173833501130 \sqrt{5}-115 i \sqrt{8669126517449449573024696780490669721303755676965485-3876911040311425488549116190343499942853937801760782 \sqrt{5}}-16557825695384603350011256367\right)^{2/5} \left(\frac{2680905286401673045006106350407138389 \sqrt{5}+6020215320576886946232982804028881471}{3660396573001917381092228630973666715067607040}+\frac{i \sqrt{8669126517449449573024696780490669721303755676965485-3876911040311425488549116190343499942853937801760782 \sqrt{5}} \left(50021450333620363744415187 \sqrt{5}+112166606653350738362971117\right) \left(94118720118901230891085096317849939 \sqrt{5}+289847623426031299198396552731049130\right)}{64624466885202183079915011863734625021224068090281978641398456898589021484851301126560268605440}\right)-\frac{209341}{10}\]
nyy 发表于 2024-1-12 16:26
让我来求解一下这个问题！
在下面网址输入
http://magma.maths.usyd.edu.au/calc/
接上面的代码：
(*对根式解求数值,再去掉"小数"部分,再排序,再合并,得到五个根*)
dd=Union@Sort@Chop@N
ee=x/.NSolve(*求解原方程的数值解*)
dd-ee(*计算解析解与数值解的差距*)
这个是解析解的数值解
{-401258.9880266900537345442003488619633104060138474772285428877281892826705713234918961527720722599939,-159268.3896774913309134341908272412846112945591483628861028830711507216950502814511642301145850513672,-110135.65894174419437008683553902302312976283880843638621954406734993046474082604111938958576053097262,174303.7437959547734151900821965408705449321758356340110574538326851764505465889277405265529998452051,391688.7928499708056028751445185854005065312359686424898078610340047583798158420564392459194179971287}
这个是由原方程求解到的数值解
{-401258.98802669005373,-159268.38967749133091,-110135.65894174419437,174303.74379595477342,391688.79284997080560}
这个是两者的差，误差很小
{0.*10^-15,0.*10^-15,0.*10^-15,0.*10^-15,0.*10^-15}
上面证明得到了所有的根式解析解，只不过解确实比较丑！
nyy 发表于 2024-1-12 16:39
接上面的代码：
Clear["Global`*"];(*Clear all variables*)
(*定义多项式,这个多项式的伽罗瓦群是可解的*)
f=32*x^5+3349456*x^4-5941616812296*x^3-585145514845851080*x^2+147013447513276833423286*x+15377302441624829616294559439;
aa={Factor,#}&/@Prime@Range;
MatrixForm
g=x^5-2*x+12;(*这个多项式没有根式解,mod p后会出现二次多项式乘以三次多项式*)
MatrixForm[{Factor,#}&/@Prime@Range];
f=32*x^5+3349456*x^4-5941616812296*x^3-585145514845851080*x^2+147013447513276833423286*x+15377302441624829616294559439;
求解结果
\[\left(
\begin{array}{cc}
1 & 2 \\
2 \left(x^5+2 x^4+2 x^2+2 x+1\right) & 3 \\
2 \left(x^5+3 x^4+2 x^3+3 x+2\right) & 5 \\
4 \left(x^5+3 x^4+6 x^3+3 x^2+4 x+3\right) & 7 \\
10 x^5 & 11 \\
6 \left(x^5+x^4+5 x^3+8 x^2+11 x+5\right) & 13 \\
15 \left(x^5+10 x^4+13 x^3+3 x^2+8\right) & 17 \\
13 \left(x^5+9 x^4+17 x^3+5 x^2+x+11\right) & 19 \\
9 x^4 (x+9) & 23 \\
3 \left(x^5+24 x^4+19 x^3+20 x^2+10 x+24\right) & 29 \\
x^5+30 x^4+26 x^3+13 x^2+x+7 & 31 \\
32 \left(x^5+16 x^4+16 x^3+12 x^2+34 x+11\right) & 37 \\
32 \left(x^5+18 x^4+17 x^3+27 x^2+3 x+31\right) & 41 \\
32 (x+13) (x+16) (x+25) (x+28) (x+34) & 43 \\
32 \left(x^5+25 x^4+22 x^3+34 x^2+27 x+2\right) & 47 \\
32 \left(x^5+22 x^4+29 x^3+44 x^2+x+42\right) & 53 \\
32 \left(x^5+34 x^4+37 x^3+49 x^2+24 x+37\right) & 59 \\
32 \left(x^5+25 x^4+23 x^3+52 x^2+38 x+39\right) & 61 \\
32 (x+29) (x+46) (x+47) (x+63) (x+66) & 67 \\
32 \left(x^5+52 x^4+64 x^3+27 x^2+5 x+19\right) & 71 \\
\end{array}
\right)\]
上面的结果没出现二次多项式乘以三次多项式，因此有根式解
对于\(x^5-2 x+12\)
\[\left(
\begin{array}{cc}
x^5 & 2 \\
x \left(x^2+x+2\right) \left(x^2+2 x+2\right) & 3 \\
(x+3) \left(x^4+2 x^3+4 x^2+3 x+4\right) & 5 \\
x^5+5 x+5 & 7 \\
(x+10) \left(x^2+5 x+3\right) \left(x^2+7 x+7\right) & 11 \\
(x+1) \left(x^2+2 x+3\right) \left(x^2+10 x+4\right) & 13 \\
(x+6) \left(x^4+11 x^3+2 x^2+5 x+2\right) & 17 \\
(x+6) \left(x^4+13 x^3+17 x^2+12 x+2\right) & 19 \\
x^5+21 x+12 & 23 \\
(x+5) (x+17) (x+19) \left(x^2+17 x+18\right) & 29 \\
x^5+29 x+12 & 31 \\
\left(x^2+10 x+27\right) \left(x^3+27 x^2+36 x+21\right) & 37 \\
(x+13)^2 \left(x^3+15 x^2+15 x+27\right) & 41 \\
x^5+41 x+12 & 43 \\
(x+20) \left(x^4+27 x^3+24 x^2+37 x+10\right) & 47 \\
(x+48) \left(x^4+5 x^3+25 x^2+19 x+40\right) & 53 \\
(x+28) (x+42) (x+54) \left(x^2+53 x+36\right) & 59 \\
(x+35) \left(x^2+12 x+6\right) \left(x^2+14 x+14\right) & 61 \\
\left(x^2+65 x+48\right) \left(x^3+2 x^2+23 x+17\right) & 67 \\
(x+10) \left(x^2+65 x+12\right) \left(x^2+67 x+64\right) & 71 \\
\end{array}
\right)\]
上面的结果出现了二次多项式乘以三次多项式
\[\left(
\begin{array}{cc}
\left(x^2+10 x+27\right) \left(x^3+27 x^2+36 x+21\right) & 37 \\
\end{array}
\right)\]
因此\(x^5-2 x+12\)无根式解
参考资料
已知首一系数的五次方程的五个系数，能确定方程的伽罗瓦群吗？ - 酱紫君的回答 - 知乎
https://www.zhihu.com/question/64900539/answer/2476505448
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 ...
看完整循环、看对换！