"Source: Text/Ring/RngLoc.text";
"Line: 1267";
"Date: Fri Sep 26 14:29:19 2025";
"Main: Fri Sep 26 14:29:19 2025";
// original file: Text/Ring/RngLoc.text, line: 1267
// Example: H49E7 ()
print "Example: H49E7";
ei := GetEchoInput();
SetEchoInput(true);
K := ext<pAdicField(2,20)|Polynomial([10,0,1])>; K;
RootNumber(K);
K := pAdicField(3,40);
A := AllExtensions(K,3);
_<x> := PolynomialRing(Integers(K)); // printing
[<RootNumber(FieldOfFractions(a)),DefiningPolynomial(a)> : a in A];
K := pAdicField(3,20);
L := ext<K|Polynomial([3,0,1])>; // both ramified
M := ext<L|Polynomial([L.1,0,1])>; // both ramified
RootNumber(M); // relative for M/L
AbsoluteRootNumber(M); // absolute for M/Q3
RootNumber(L); // intermediate field
X := AbsoluteTotallyRamifiedExtension(Integers(M));
RootNumber(FieldOfFractions(X));
SetEchoInput(ei);
