"Source: Text/Algebra/AlgFP.text";
"Line: 1770";
"Date: Thu Sep 25 14:47:46 2025";
"Main: Fri Sep 26 14:29:19 2025";
// original file: Text/Algebra/AlgFP.text, line: 1770
// Example: H91E7 ()
print "Example: H91E7";
ei := GetEchoInput();
SetEchoInput(true);
K := RationalField();
A<x,y,z> := FPAlgebra<K, x,y,z |
          x^2 - y*z*y + z, y^2 - y*x*y + 1, z^2 - y*x*y - x*z*x>;
A;
IsCommutative(A);
assert not $1;
y*z;
z*y;
U<u> := PolynomialRing(K);
MinimalPolynomial(x);
MinimalPolynomial(y);
Dimension(A);
assert $1 eq 18;
V, Vf := VectorSpace(A);
V;
Vf(x);
[V.i@@Vf: i in [1 .. Dimension(V)]];
M, Mf := MatrixAlgebra(A);
M;
M.1;
M.1 eq RepresentationMatrix(x);
assert $1;
MinimalPolynomial(M.1);        
FactoredMinimalPolynomial(M.1);
N := Kernel(M.1); 
N;
a := N.1 @@ Vf;
a;
IsNilpotent(a);
a^3;
assert $1 eq 0;
S, Sf := Algebra(A);
S;
Centre(S);
J := JacobsonRadical(S);
J;
L := [J.i@@Sf: i in [1 .. 4]];
L;
[MinimalPolynomial(x): x in L];
SetEchoInput(ei);
