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gap> 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)]];
[ [ 0*Z(5), 0*Z(5), 0*Z(5), Z(5^2)^3 ], [ 0*Z(5), 0*Z(5), Z(5^2)^3, 0*Z(5) ],
[ 0*Z(5), Z(5^2)^15, 0*Z(5), 0*Z(5) ],
[ Z(5^2)^15, 0*Z(5), 0*Z(5), 0*Z(5) ] ]
gap> form := HermitianFormByMatrix(mat,GF(25));
< hermitian form >
gap> Display(form);
Hermitian form
Gram Matrix:
z = Z(25)
. . . z^3
. . z^3 .
. z^15 . .
z^15 . . .
gap> WittIndex(form);
2
gap> form2 := BilinearFormByMatrix(mat,GF(25));
< bilinear form >
gap> Display(form2);
Bilinear form
Gram Matrix:
z = Z(25)
. . . z^3
. . z^3 .
. z^15 . .
z^15 . . .
gap> IsAlternatingForm(form2);
true
gap> Display(IsometricCanonicalForm(form));
Hermitian form
Gram Matrix:
1 . . .
. 1 . .
. . 1 .
. . . 1
Witt Index: 2
gap> Display(IsometricCanonicalForm(form2));
Bilinear form
Gram Matrix:
. 1 . .
4 . . .
. . . 1
. . 4 .
Witt Index: 2
gap> V := GF(25)^4;
( GF(5^2)^4 )
gap> u := [Z(5)^0,Z(5^2)^11,Z(5)^3,Z(5^2)^13 ];
[ Z(5)^0, Z(5^2)^11, Z(5)^3, Z(5^2)^13 ]
gap> [u,u]^form;
0*Z(5)
gap> v := [Z(5)^0,Z(5^2)^5,Z(5^2),Z(5^2)^13 ];
[ Z(5)^0, Z(5^2)^5, Z(5^2), Z(5^2)^13 ]
gap> [v,v]^form;
0*Z(5)
gap> [u,v]^form;
Z(5^2)^7
gap> ([v,u]^form)^5;
Z(5^2)^7
gap> w := [Z(5^2)^21,Z(5^2)^19,Z(5^2)^4,Z(5)^3 ];
[ Z(5^2)^21, Z(5^2)^19, Z(5^2)^4, Z(5)^3 ]
gap> [w,w]^form;
Z(5)
gap> v := [Z(5)^0,Z(5^2)^10,Z(5^2)^15,Z(5^2)^3 ];
[ Z(5)^0, Z(5^2)^10, Z(5^2)^15, Z(5^2)^3 ]
gap> u := [Z(5)^3,Z(5^2)^9,Z(5^2)^4,Z(5^2)^16 ];
[ Z(5)^3, Z(5^2)^9, Z(5^2)^4, Z(5^2)^16 ]
gap> w := [Z(5)^2,Z(5^2)^9,Z(5^2)^23,Z(5^2)^11 ];
[ Z(5)^2, Z(5^2)^9, Z(5^2)^23, Z(5^2)^11 ]
gap> [u,v]^form;
0*Z(5)
gap> [u,w]^form;
0*Z(5)
gap> [v,w]^form;
0*Z(5)
gap> s := Subspace(V,[v,u,w]);
<vector space over GF(5^2), with 3 generators>
gap> Dimension(s);
2
gap> WittIndex(form);
2
[ Dauer der Verarbeitung: 0.14 Sekunden
(vorverarbeitet)
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