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#Background theory: example 5
mat := [[0*Z(5),0*Z(5),0*Z(25),Z(25)^3],[0*Z(5),0*Z(5),Z(25)^3,0*Z(25)],
[0*Z(5),-Z(25)^3,0*Z(5),0*Z(5)],[-Z(25)^3,0*Z(5),0*Z(25),0*Z(25)]];
form := HermitianFormByMatrix(mat,GF(25));
Display(form);
WittIndex(form);
form2 := BilinearFormByMatrix(mat,GF(25));
Display(form2);
IsAlternatingForm(form2);
Display(IsometricCanonicalForm(form));
Display(IsometricCanonicalForm(form2));
V := GF(25)^4;
u := [Z(5)^0,Z(5^2)^11,Z(5)^3,Z(5^2)^13 ];
[u,u]^form;
v := [Z(5)^0,Z(5^2)^5,Z(5^2),Z(5^2)^13 ];
[v,v]^form;
[u,v]^form;
([v,u]^form)^5;
w := [Z(5^2)^21,Z(5^2)^19,Z(5^2)^4,Z(5)^3 ];
[w,w]^form;
v := [Z(5)^0,Z(5^2)^10,Z(5^2)^15,Z(5^2)^3 ];
u := [Z(5)^3,Z(5^2)^9,Z(5^2)^4,Z(5^2)^16 ];
w := [Z(5)^2,Z(5^2)^9,Z(5^2)^23,Z(5^2)^11 ];
[u,v]^form;
[u,w]^form;
[v,w]^form;
s := Subspace(V,[v,u,w]);
Dimension(s);
WittIndex(form);
quit;
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