"Source: Text/Ring/FldFunG.text";
"Line: 1770";
"Date: Fri Sep 26 13:15:42 2025";
"Main: Fri Sep 26 14:29:19 2025";
// original file: Text/Ring/FldFunG.text, line: 1770
// Example: H46E14 ()
print "Example: H46E14";
ei := GetEchoInput();
SetEchoInput(true);
k<x>:= FunctionField(GF(1009));
R<y>:= PolynomialRing(k);
f:= y^10 + (989*x^4 + 20*x^3 + 989*x^2 + 20*x + 989)*y^8 + (70*x^8 + 
869*x^7 + 310*x^6 + 529*x^5 + 600*x^4 + 479*x^3 + 460*x^2 + 719*x + 
120)*y^6 + (909*x^12 + 300*x^11 + 409*x^10 + 1000*x^9 + 393*x^8 + 
657*x^7 + 895*x^6 + 764*x^5 + 420*x^4 + 973*x^3 + 177*x^2 + 166*x + 
784)*y^4 + (65*x^16 + 749*x^15 + 350*x^14 + 909*x^13 + 484*x^12 + 
452*x^11 + 115*x^10 + 923*x^9 + 541*x^8 + 272*x^7 + 637*x^6 + 314*x^5 + 
724*x^4 + 490*x^3 + 948*x^2 + 99*x + 90)*y^2 + 993*x^20 + 80*x^19 + 
969*x^18 + 569*x^17 + 895*x^16 + 101*x^15 + 742*x^14 + 587*x^13 + 
55*x^12+ 437*x^11 + 97*x^10 + 976*x^9 + 62*x^8 + 171*x^7 + 930*x^6 + 
604*x^5 + 698*x^4 + 60*x^3 + 60*x^2 + 1004*x + 1008;
G, r, p:= GaloisGroup(f);
t1, t2:= TransitiveGroupIdentification(G);
t1;
assert $1 eq 1;
t2;
assert $1 eq 10;
k<t>:= FunctionField(GF(7));
R<x>:= PolynomialRing(k);
f:= x^12 + x^10 + x^8 + (6*t^2 + 3)*x^6 + (4*t^4 + 6*t^2 + 1)*x^4 + 
(5*t^4 + t^2)*x^2 + 2*t^4;
G, r, p:= GaloisGroup(f);
G;
A := Universe(r);
AssignNames(~A,  ["t"]);
_<w> := CoefficientRing(Universe(r));
_<u> := CoefficientField(Parent(w));
r;
p;
p`Prime;
p`Ring;
SetEchoInput(ei);
