"Source: Text/Ring/RngLoc.text";
"Line: 4129";
"Date: Fri Sep 26 14:29:19 2025";
"Main: Fri Sep 26 14:29:19 2025";
// original file: Text/Ring/RngLoc.text, line: 4129
// Example: H49E29 ()
print "Example: H49E29";
ei := GetEchoInput();
SetEchoInput(true);
R := pAdicRing(3,20);
_<x> := PolynomialRing(R); // for printing
NumberOfExtensions(R,9 : F:=1,j:=4);
assert $1 eq 54;
A9 := AllExtensions(R,9 : F:=1,j:=4);
[DefiningPolynomial(a) : a in A9];
A3 := AllExtensions(R,3 : F:=1,j:=1);
NumberOfExtensions(A3[1],3 : F:=1,j:=1);
assert $1 eq 6;
[DefiningPolynomial(a) : a in A3];
_<pi> := A3[1];
_<y> := PolynomialRing(A3[1]);
B3 := AllExtensions(A3[1],3 : F:=1,j:=1);
[DefiningPolynomial(f) : f in B3];
R := pAdicRing(5,20);
NumberOfExtensions(R,10);
assert $1 eq 1818;
[NumberOfExtensions(R,10 : E:=e) : e in Divisors(10)];
assert $1 eq [1,2,605,1210];
U := UnramifiedExtension(R,2);
[#AllExtensions(U,5 : E:=5,j:=j0) : j0 in [1..5]];
assert $1 eq [24,24,24,48,25];
// compare the above/below to (#K)/N in Pauli-Roblot
[#AllExtensions(R,10 : E:=10,j:=j0): j0 in [1..10]];
assert $1 eq [8,8,8,16,0,40,40,80,40,50];
SetEchoInput(ei);
