求x^5+x+12的伽罗瓦群，子群，正规子群，合成列，生成元，代码
// 计算多项式 x^5 + x + 12 的伽罗瓦群P<x> := PolynomialRing(RationalField());
f := x^5 + x + 12;
print "Polynomial:", f;
G := GaloisGroup(f);
print "Galois group:", G;
print "Order:", #G;
print "Isomorphic to:", GroupName(G);
print "-----------------------------------";
// 所有子群
Subs := Subgroups(G);
print "All subgroups (by order):";
for sub in Subs do
print "Order", #sub`subgroup, ":", GroupName(sub`subgroup);
end for;
print "-----------------------------------";
// 正规子群
NormalSubs := NormalSubgroups(G);
print "Normal subgroups:";
for nsub in NormalSubs do
print "Order", #nsub`subgroup, ":", GroupName(nsub`subgroup);
end for;
print "-----------------------------------";
// 合成列
Comp := CompositionSeries(G);
print "Composition series:";
for i in do
print i, ":", GroupName(Comp);
end for;
print "-----------------------------------";
// 生成元
print "Generators of Galois group:";
print Generators(G);
print "-----------------------------------";
// 判断是否可解
print "Is solvable?", IsSolvable(G); 本帖最后由 nyy 于 2025-8-27 10:27 编辑
Polynomial: x^5 + x + 12
Galois group: Symmetric group G acting on a set of cardinality 5
Order = 120 = 2^3 * 3 * 5
Order: 120
Isomorphic to: S5
-----------------------------------
All subgroups (by order):
Order 1 : C1
Order 2 : C2
Order 2 : C2
Order 3 : C3
Order 4 : C4
Order 4 : C2^2
Order 4 : C2^2
Order 5 : C5
Order 6 : S3
Order 6 : C6
Order 6 : S3
Order 8 : D4
Order 10 : D5
Order 12 : A4
Order 12 : D6
Order 20 : F5
Order 24 : S4
Order 60 : A5
Order 120 : S5
-----------------------------------
Normal subgroups:
Order 1 : C1
Order 60 : A5
Order 120 : S5
-----------------------------------
Composition series:
1 : S5
2 : A5
3 : C1
-----------------------------------
Generators of Galois group:
{
(1, 2),
(1, 2, 3, 4, 5)
}
-----------------------------------
Is solvable? false
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。 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/
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