[例1]f0,f1,f2,f3のガロア群

_<x>:=PolynomialAlgebra(Rationals());

f0:=x^5+1;
f1:=x^5-10*x^3+5*x^2+10*x+1;
f2:=x^5-5*x+12;
f3:=x^5+15*x+12;

f0,GaloisGroup(f0);
f1,GaloisGroup(f1);
f2,GaloisGroup(f2);
f3,GaloisGroup(f3);


[例2]f(x)=x^5+ax^3+bx^2+cx+d(-1<a,b,c<3, 0<d<5)のガロア群

_<x>:=PolynomialAlgebra(Rationals());

for i in {0..3} do
 for j in {0..3} do
  for k in {0..3} do
   for m in {1..5} do
     f:=x^5+i*x^3+j*x^2+k*x+m;
     g:=GaloisGroup(f);
     if Order(g) le 60 then 
       print f, g;
     end if;
   end for;
  end for;
 end for;
end for;


[例3]f=x^3-2 の原始元vの最小多項式と，vによるfの解の表現
R<x>:=PolynomialRing(Rationals());
f:=x^3-2;
K<s>:=NumberField(f);
Degree(K);

S<v>:=SplittingField(x^3-2);
S;
fn:=Factorization(Polynomial(S,x^3-2));
      fn;
C:=[-Coefficient(g[1],0): g in fn];
C;

[例4]10次方程式のGalois群
_<x>:=PolynomialAlgebra(Rationals());
f:=x^10-5*x^5+9;
g:=GaloisGroup(f);g;

[例5]100次方程式のGalois群
_<x>:=PolynomialAlgebra(Rationals());
f:=x^100-3*x+1;
g:=GaloisGroup(f);
g;