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[分享] 五次方程、伽罗瓦群、群论等相关问题

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nyy
 楼主| 发表于 2024-1-17 09:04:53 | 显示全部楼层
polgalois(x^5 + x^4 - 4*x^3 - 3*x^2 + 3*x + 1)
这个群是C5，属于可解的，有根式解
毋因群疑而阻独见　　毋任己意而废人言
毋私小惠而伤大体　　毋借公论以快私情
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nyy
 楼主| 发表于 2024-1-17 09:09:22 | 显示全部楼层
https://math.stackexchange.com/q ... quintic-in-radicals

How to solve a cyclic quintic in radicals?

  1. NB: I generated this (in a gap4 session, with RadiRoot properly installed) as follows:

  3. gap> LoadPackage("radiroot");
  4. ---------------------------------------------------------------------------------
  5. Loading  RadiRoot 2.4 (Roots of a Polynomial as Radicals)
  6. by Andreas Distler (a.distler@tu-bs.de).
  7. ---------------------------------------------------------------------------------
  8. true
  9. gap> g := UnivariatePolynomial( Rationals, [1,1,1,1,1,1,1,1,1,1,1]);
  10. x_1^10+x_1^9+x_1^8+x_1^7+x_1^6+x_1^5+x_1^4+x_1^3+x_1^2+x_1+1
  11. gap>  RootsOfPolynomialAsRadicals(g, "latex");
  12. "/tmp/tmp.sfoZ6C/Nst.tex"
  13. gap>
  14. That created a LATEX file, /tmp/tmp.sfoZ6C/Nst.tex, containing the formulas
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毋因群疑而阻独见　　毋任己意而废人言
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nyy
 楼主| 发表于 2024-1-17 09:11:30 | 显示全部楼层
For primes p≡1(mod10), the method of Gauss gives the roots of a certain quintic as sums of ζn, where ζ=e2πi/p. In turn, the collection of exponents n is of the form wk. Example, for p=31 the exponents are 6k for 0≤k≤5. Here are the first 35 such polynomials and the exponents for one of the roots. Compare with the list for p=11,31,41,61,71 in https://en.wikipedia.org/wiki/Qu ... r_solvable_quintics

https://math.stackexchange.com/q ... x2x5-0?noredirect=1
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nyy
 楼主| 发表于 2024-1-17 11:28:30 | 显示全部楼层
本帖最后由 nyy 于 2024-1-17 11:35 编辑
  1. [-10, -5, 0, 0, -6][60, 1, 1, "A5"]
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五次方程，绝大多数都是S5，
这个是A5
方程表示x^5-10*x^4-5*x^3-6，这个多项式的伽罗瓦群是A5，因为很少，所以我记一下

  1. (11:19) gp > for(a4=-10,10,for(a3=-10,10,for(a2=-10,10,for(a1=-10,10,for(a0=-10,10,f=x^5+a4*x^4+a3*x^3+a2*x^2+a1*x+a0;if(polisirreducible(f),print([a4,a3,a2,a1,a0],polgalois(f))))))))
  2. [1,-10, -10, -5, -5, -4][20, -1, 1, "F(5) = 5:4"]
  3. [1,-10, -10, 10, 10, 2][20, -1, 1, "F(5) = 5:4"]
  4. [1,-10, -8, -8, -5, -2][10, 1, 1, "D(5) = 5:2"]
  5. [1,-10, -5, -10, -5, -3][20, -1, 1, "F(5) = 5:4"]
  6. [1,-10, -5, 0, -5, -6][20, -1, 1, "F(5) = 5:4"]
  7. [1,-10, -5, 5, 1, -1][10, 1, 1, "D(5) = 5:2"]
  8. [1,-10, -5, 10, 5, -5][20, -1, 1, "F(5) = 5:4"]
  9. [1,-10, -2, -8, -1, -2][20, -1, 1, "F(5) = 5:4"]
  10. [1,-10, 0, -10, 0, -2][20, -1, 1, "F(5) = 5:4"]
  11. [1,-10, 0, 10, 0, -2][20, -1, 1, "F(5) = 5:4"]
  12. [1,-10, 3, -4, -1, -6][10, 1, 1, "D(5) = 5:2"]
  13. [1,-10, 5, 0, -10, -8][10, 1, 1, "D(5) = 5:2"]
  14. [1,-10, 5, 4, 3, -4][20, -1, 1, "F(5) = 5:4"]
  15. [1,-10, 5, 10, 0, -1][5, 1, 1, "C(5) = 5"]
  16. [1,-10, 9, -4, 0, -1][10, 1, 1, "D(5) = 5:2"]
  17. [1,-10, 10, -5, 4, -4][10, 1, 1, "D(5) = 5:2"]
  18. [1,-10, 10, -5, 5, -10][20, -1, 1, "F(5) = 5:4"]
  19. [1,-10, 10, 0, 0, -4][20, -1, 1, "F(5) = 5:4"]
  20. [1,-10, 10, 10, -10, 2][20, -1, 1, "F(5) = 5:4"]
  21. [1,-9, -5, 9, 4, -1][5, 1, 1, "C(5) = 5"]
  22. [1,-9, -4, 1, -4, -5][20, -1, 1, "F(5) = 5:4"]
  23. [1,-9, -4, 9, -5, -9][10, 1, 1, "D(5) = 5:2"]
  24. [1,-9, -2, -6, -3, -2][20, -1, 1, "F(5) = 5:4"]
  25. [1,-9, -2, 3, 7, -4][10, 1, 1, "D(5) = 5:2"]
  26. [1,-9, 3, -3, -1, -5][10, 1, 1, "D(5) = 5:2"]
  27. [1,-9, 8, -7, 3, -1][10, 1, 1, "D(5) = 5:2"]
  28. [1,-9, 9, -9, 5, -1][10, 1, 1, "D(5) = 5:2"]
  29. [1,-9, 9, 1, -1, -3][20, -1, 1, "F(5) = 5:4"]
  30. [1,-8, -6, -8, 2, -6][20, -1, 1, "F(5) = 5:4"]
  31. [1,-8, -6, -3, -1, -2][20, -1, 1, "F(5) = 5:4"]
  32. [1,-8, -4, 3, 0, -1][10, 1, 1, "D(5) = 5:2"]
  33. [1,-8, -3, 2, -3, -4][20, -1, 1, "F(5) = 5:4"]
  34. [1,-8, -1, -8, 6, -1][10, 1, 1, "D(5) = 5:2"]
  35. [1,-8, 2, -4, 1, -4][20, -1, 1, "F(5) = 5:4"]
  36. [1,-8, 3, -2, -1, -4][10, 1, 1, "D(5) = 5:2"]
  37. [1,-8, 6, 3, -6, -4][10, 1, 1, "D(5) = 5:2"]
  38. [1,-8, 7, -3, 1, -5][10, 1, 1, "D(5) = 5:2"]
  39. [1,-8, 8, 2, -2, -2][20, -1, 1, "F(5) = 5:4"]
  40. [1,-7, -9, -8, 0, -8][20, -1, 1, "F(5) = 5:4"]
复制代码
毋因群疑而阻独见　　毋任己意而废人言
毋私小惠而伤大体　　毋借公论以快私情
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nyy
 楼主| 发表于 2024-1-17 12:03:23 | 显示全部楼层
nyy 发表于 2024-1-17 11:28
五次方程，绝大多数都是S5，
这个是A5
方程表示x^5-10*x^4-5*x^3-6，这个多项式的伽罗瓦群是A5，因为很少 ...
  1. /*选择多项式函数,穷举法,20240117_120150*/
  2. {
  3.     for(a4=0,0,
  4.         for(a3=-1,1,
  5.             for(a2=-10,10,
  6.                 for(a1=-10,10,
  7.                     for(a0=-10,10,
  8.                         f=x^5+a4*x^4+a3*x^3+a2*x^2+a1*x+a0;/*定义多项式*/
  9.                         if(polisirreducible(f), /*如果是不可约多项式*/
  10.                             ga=polgalois(f);/*得到伽罗瓦群相关信息*/
  11.                             if(ga[1]<120,
  12.                                 print([1,a4,a3,a2,a1,a0],ga) /*打印多项式系数与伽罗瓦群信息*/
  13.                             )
  14.                         )
  15.                     )
  16.                 )
  17.             )
  18.         )
  19.     )
  20. }
复制代码


这个是parigp的代码，大括号是必须的！

输出结果
  1.                   GP/PARI CALCULATOR Version 2.15.4 (released)
  2.           amd64 running mingw (x86-64/GMP-6.1.2 kernel) 64-bit version
  3.            compiled: Jun 28 2023, gcc version 10-posix 20210110 (GCC)
  4.                             threading engine: single
  5.                  (readline v8.0 enabled, extended help enabled)

  7.                      Copyright (C) 2000-2022 The PARI Group

  9. PARI/GP is free software, covered by the GNU General Public License, and comes
  10. WITHOUT ANY WARRANTY WHATSOEVER.

  12. Type ? for help, \q to quit.
  13. Type ?18 for how to get moral (and possibly technical) support.

  15. parisizemax = 400000000, primelimit = 500000
  16. (12:00) gp > \r"C:\Program Files (x86)\Pari64-2-15-4\my_script\selectpol.gp"
  17. [1, 0, -1, -10, 8, 9][10, 1, 1, "D(5) = 5:2"]
  18. [1, 0, -1, -7, -1, -3][10, 1, 1, "D(5) = 5:2"]
  19. [1, 0, -1, -7, 8, -5][60, 1, 1, "A5"]
  20. [1, 0, -1, -4, 4, -1][10, 1, 1, "D(5) = 5:2"]
  21. [1, 0, -1, -4, 7, 2][20, -1, 1, "F(5) = 5:4"]
  22. [1, 0, -1, -4, 10, -4][10, 1, 1, "D(5) = 5:2"]
  23. [1, 0, -1, -2, -2, -1][10, 1, 1, "D(5) = 5:2"]
  24. [1, 0, -1, -2, 3, 4][10, 1, 1, "D(5) = 5:2"]
  25. [1, 0, -1, 2, -2, 1][10, 1, 1, "D(5) = 5:2"]
  26. [1, 0, -1, 2, 3, -4][10, 1, 1, "D(5) = 5:2"]
  27. [1, 0, -1, 4, 4, 1][10, 1, 1, "D(5) = 5:2"]
  28. [1, 0, -1, 4, 7, -2][20, -1, 1, "F(5) = 5:4"]
  29. [1, 0, -1, 4, 10, 4][10, 1, 1, "D(5) = 5:2"]
  30. [1, 0, -1, 7, -1, 3][10, 1, 1, "D(5) = 5:2"]
  31. [1, 0, -1, 7, 8, 5][60, 1, 1, "A5"]
  32. [1, 0, -1, 10, 8, -9][10, 1, 1, "D(5) = 5:2"]
  33. [1, 0, 0, -10, -10, -6][20, -1, 1, "F(5) = 5:4"]
  34. [1, 0, 0, -10, 10, 2][20, -1, 1, "F(5) = 5:4"]
  35. [1, 0, 0, -5, -5, -8][20, -1, 1, "F(5) = 5:4"]
  36. [1, 0, 0, -5, -4, -1][10, 1, 1, "D(5) = 5:2"]
  37. [1, 0, 0, -5, 0, -3][10, 1, 1, "D(5) = 5:2"]
  38. [1, 0, 0, -5, 10, -4][10, 1, 1, "D(5) = 5:2"]
  39. [1, 0, 0, -4, 1, 4][60, 1, 1, "A5"]
  40. [1, 0, 0, -3, 2, 1][10, 1, 1, "D(5) = 5:2"]
  41. [1, 0, 0, -1, -2, -3][60, 1, 1, "A5"]
  42. [1, 0, 0, 0, 0, -10][20, -1, 1, "F(5) = 5:4"]
  43. [1, 0, 0, 0, 0, -9][20, -1, 1, "F(5) = 5:4"]
  44. [1, 0, 0, 0, 0, -8][20, -1, 1, "F(5) = 5:4"]
  45. [1, 0, 0, 0, 0, -7][20, -1, 1, "F(5) = 5:4"]
  46. [1, 0, 0, 0, 0, -6][20, -1, 1, "F(5) = 5:4"]
  47. [1, 0, 0, 0, 0, -5][20, -1, 1, "F(5) = 5:4"]
  48. [1, 0, 0, 0, 0, -4][20, -1, 1, "F(5) = 5:4"]
  49. [1, 0, 0, 0, 0, -3][20, -1, 1, "F(5) = 5:4"]
  50. [1, 0, 0, 0, 0, -2][20, -1, 1, "F(5) = 5:4"]
  51. [1, 0, 0, 0, 0, 2][20, -1, 1, "F(5) = 5:4"]
  52. [1, 0, 0, 0, 0, 3][20, -1, 1, "F(5) = 5:4"]
  53. [1, 0, 0, 0, 0, 4][20, -1, 1, "F(5) = 5:4"]
  54. [1, 0, 0, 0, 0, 5][20, -1, 1, "F(5) = 5:4"]
  55. [1, 0, 0, 0, 0, 6][20, -1, 1, "F(5) = 5:4"]
  56. [1, 0, 0, 0, 0, 7][20, -1, 1, "F(5) = 5:4"]
  57. [1, 0, 0, 0, 0, 8][20, -1, 1, "F(5) = 5:4"]
  58. [1, 0, 0, 0, 0, 9][20, -1, 1, "F(5) = 5:4"]
  59. [1, 0, 0, 0, 0, 10][20, -1, 1, "F(5) = 5:4"]
  60. [1, 0, 0, 1, -2, 3][60, 1, 1, "A5"]
  61. [1, 0, 0, 3, 2, -1][10, 1, 1, "D(5) = 5:2"]
  62. [1, 0, 0, 4, 1, -4][60, 1, 1, "A5"]
  63. [1, 0, 0, 5, -5, 8][20, -1, 1, "F(5) = 5:4"]
  64. [1, 0, 0, 5, -4, 1][10, 1, 1, "D(5) = 5:2"]
  65. [1, 0, 0, 5, 0, 3][10, 1, 1, "D(5) = 5:2"]
  66. [1, 0, 0, 5, 10, 4][10, 1, 1, "D(5) = 5:2"]
  67. [1, 0, 0, 10, -10, 6][20, -1, 1, "F(5) = 5:4"]
  68. [1, 0, 0, 10, 10, -2][20, -1, 1, "F(5) = 5:4"]
  69. [1, 0, 1, -6, 4, -3][10, 1, 1, "D(5) = 5:2"]
  70. [1, 0, 1, -6, 10, -4][10, 1, 1, "D(5) = 5:2"]
  71. [1, 0, 1, -5, -4, -7][60, 1, 1, "A5"]
  72. [1, 0, 1, -3, 0, 3][60, 1, 1, "A5"]
  73. [1, 0, 1, -3, 1, -3][10, 1, 1, "D(5) = 5:2"]
  74. [1, 0, 1, -1, 6, -8][60, 1, 1, "A5"]
  75. [1, 0, 1, 1, 6, 8][60, 1, 1, "A5"]
  76. [1, 0, 1, 3, 0, -3][60, 1, 1, "A5"]
  77. [1, 0, 1, 3, 1, 3][10, 1, 1, "D(5) = 5:2"]
  78. [1, 0, 1, 5, -4, 7][60, 1, 1, "A5"]
  79. [1, 0, 1, 6, 4, 3][10, 1, 1, "D(5) = 5:2"]
  80. [1, 0, 1, 6, 10, 4][10, 1, 1, "D(5) = 5:2"]
复制代码

毋因群疑而阻独见　　毋任己意而废人言
毋私小惠而伤大体　　毋借公论以快私情
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nyy
 楼主| 发表于 2024-1-18 11:31:02 | 显示全部楼层
对称群S6的所有子群
https://zhuanlan.zhihu.com/p/668078206
毋因群疑而阻独见　　毋任己意而废人言
毋私小惠而伤大体　　毋借公论以快私情
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nyy
 楼主| 发表于 2024-1-22 10:04:55 | 显示全部楼层
Galois准则的另一个结果是不可约可解n次方程的Galois群的大小始终是n(n−1)的约数和n的倍数。

【机翻】 方程的Galois群 - 方汝见之的文章 - 知乎
https://zhuanlan.zhihu.com/p/661077428
毋因群疑而阻独见　　毋任己意而废人言
毋私小惠而伤大体　　毋借公论以快私情
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nyy
 楼主| 发表于 2024-1-22 10:30:01 | 显示全部楼层
https://www-users.cse.umn.edu/~garrett/m/algebra/notes/23.pdf
23. Solving equations by radicals
23.1 Galois’ criterion
23.2 Composition series, Jordan-H¨older theorem
23.3 Solving cubics by radicals
23.4 Worked examples
Around 1800, Ruffini sketched a proof, completed by Abel, that the general quintic equation is not solvable in
radicals, by contrast to cubics and quartics whose solutions by radicals were found in the Italian renaissance,
not to mention quadratic equations, understood in antiquity. Ruffini’s proof required classifying the possible
forms of radicals. By contrast, Galois’ systematic development of the idea of automorphism group replaced
the study of the expressions themselves with the study of their movements.
Galois theory solves some classical problems. Ruler-and-compass constructions, in coordinates, can only
express quantities in repeated quadratic extensions of the field generated by given points, but nothing else.
Thus, trisection of angles by ruler and compass is impossible for general-position angles, since the general
trisection requires a cube root.
The examples and exercises continue with other themes
毋因群疑而阻独见　　毋任己意而废人言
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 楼主| 发表于 2024-1-22 10:37:15 | 显示全部楼层
https://mathsci.kaist.ac.kr/~jkoenig/Lineout_Talk_Computation.pdf
Methods for computing Galois groups
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