"Source: Text/Ring/RngSer.text";
"Line: 2125";
"Date: Fri Sep 26 12:10:11 2025";
"Main: Fri Sep 26 14:29:19 2025";
// original file: Text/Ring/RngSer.text, line: 2125
// Example: H51E7 ()
print "Example: H51E7";
ei := GetEchoInput();
SetEchoInput(true);
P<t> := PowerSeriesRing(GF(101), 50);
R<x> := PolynomialRing(P);
U<u> := UnramifiedExtension(P, x^2 + 2*x + 3);
UF := FieldOfFractions(U);
R<y> := PolynomialRing(U);
T<tt> := TotallyRamifiedExtension(U, y^2 + t*y + t);
TF<tf> := FieldOfFractions(T);
UF<uf> := FieldOfFractions(U);                      
u + t;
uf * t;
tf eq tt;
assert not $1;
tf - tt; 
IsZero($1);
assert not $1;
Valuation($2);
assert $1 eq 100;
Valuation(tt); 
assert $1 eq 1;
Valuation(U!t);
assert $1 eq 1;
Valuation(T!t);
assert $1 eq 2;
RelativePrecision(u);
assert $1 eq 50;
AbsolutePrecision(u);
assert $1 eq 50;
AbsolutePrecision(uf);
assert $1 eq 50;
RelativePrecision(uf);
assert $1 eq 50;
RelativePrecision(u - uf);
assert $1 eq 0;
AbsolutePrecision(u - uf); 
assert $1 eq 50;
u^7;
Coefficients($1);
tt^8;
Coefficients($1);
SetEchoInput(ei);
