求x^5+x+12的伽罗瓦群，子群，正规子群，合成列，生成元，代码

nyy

1. // 计算多项式 x^5 + x + 12 的伽罗瓦群
   
2. P<x> := PolynomialRing(RationalField());
   
3. f := x^5 + x + 12;
   
4. print "Polynomial:", f;
   
5. 
   
6. G := GaloisGroup(f);
   
7. print "Galois group:", G;
   
8. print "Order:", #G;
   
9. print "Isomorphic to:", GroupName(G);
   
10. print "-----------------------------------";
    
11. 
    
12. // 所有子群
    
13. Subs := Subgroups(G);
    
14. print "All subgroups (by order):";
    
15. for sub in Subs do
    
16.     print "Order", #sub`subgroup, ":", GroupName(sub`subgroup);
    
17. end for;
    
18. print "-----------------------------------";
    
19. 
    
20. // 正规子群
    
21. NormalSubs := NormalSubgroups(G);
    
22. print "Normal subgroups:";
    
23. for nsub in NormalSubs do
    
24.     print "Order", #nsub`subgroup, ":", GroupName(nsub`subgroup);
    
25. end for;
    
26. print "-----------------------------------";
    
27. 
    
28. // 合成列
    
29. Comp := CompositionSeries(G);
    
30. print "Composition series:";
    
31. for i in [1..#Comp] do
    
32.     print i, ":", GroupName(Comp[i]);
    
33. end for;
    
34. print "-----------------------------------";
    
35. 
    
36. // 生成元
    
37. print "Generators of Galois group:";
    
38. print Generators(G);
    
39. print "-----------------------------------";
    
40. 
    
41. // 判断是否可解
    
42. print "Is solvable?", IsSolvable(G);

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nyy

1. Polynomial: x^5 + x + 12
   
2. Galois group: Symmetric group G acting on a set of cardinality 5
   
3. Order = 120 = 2^3 * 3 * 5
   
4. Order: 120
   
5. Isomorphic to: S5
   
6. -----------------------------------
   
7. All subgroups (by order):
   
8. Order 1 : C1
   
9. Order 2 : C2
   
10. Order 2 : C2
    
11. Order 3 : C3
    
12. Order 4 : C4
    
13. Order 4 : C2^2
    
14. Order 4 : C2^2
    
15. Order 5 : C5
    
16. Order 6 : S3
    
17. Order 6 : C6
    
18. Order 6 : S3
    
19. Order 8 : D4
    
20. Order 10 : D5
    
21. Order 12 : A4
    
22. Order 12 : D6
    
23. Order 20 : F5
    
24. Order 24 : S4
    
25. Order 60 : A5
    
26. Order 120 : S5
    
27. -----------------------------------
    
28. Normal subgroups:
    
29. Order 1 : C1
    
30. Order 60 : A5
    
31. Order 120 : S5
    
32. -----------------------------------
    
33. Composition series:
    
34. 1 : S5
    
35. 2 : A5
    
36. 3 : C1
    
37. -----------------------------------
    
38. Generators of Galois group:
    
39. {
    
40.     (1, 2),
    
41.     (1, 2, 3, 4, 5)
    
42. }
    
43. -----------------------------------
    
44. Is solvable? false

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求解结果，

给出这些代码与求解结果的例子，
更有利于学习抽象代数。

nyy

奇怪的是为什么子群里有两个4阶群相同？
6阶群也两个相同？
是bug？

软件magma

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