"Source: Text/Ring/FldAb.text";
"Line: 734";
"Date: Fri Sep 26 12:10:11 2025";
"Main: Fri Sep 26 14:29:19 2025";
// original file: Text/Ring/FldAb.text, line: 734
// Example: H41E5 ()
print "Example: H41E5";
ei := GetEchoInput();
SetEchoInput(true);
x := ext<Rationals()|>.1;
Q := ext<Rationals()| x-1 :DoLinearExtension>;
M := MaximalOrder(Q);
G, m := RayClassGroup(12*M, [1]);
G;
A := AbelianExtension(m);
E := EquationOrder(A);
Ea := SimpleExtension(E);
Ma := MaximalOrder(Ea);
Discriminant(Ma);
assert $1 eq 144;
Factorization(Polynomial(Ma, CyclotomicPolynomial(12)));
x := ext<Integers()|>.1;
M := MaximalOrder(x^2-10);
G, m := RayClassGroup(3615*M, [1,2]);
G; m;
h := hom<G -> G | [5*G.i : i in [1..#Generators(G)]]>;
Q, mq := quo<G|Image(h)>;
mm := Inverse(mq) * m;
mm;
A := AbelianExtension(mm);
E := EquationOrder(A);
E;
C := Components(A);
GaloisGroup(NumberField(C[1]));
GaloisGroup(NumberField(C[2]));
SetEchoInput(ei);
