GAP 4.8.9 installation with standard packages -- copy to your CoCalc project to get it

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\Chapter{An example application}
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In this section we outline two example computations with the functions
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of the previous chapter. The first example uses number fields defined 
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by matrices and the second example considers number fields defined by
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a polynomial.
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\Section{Number fields defined by matrices}
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\beginexample
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gap> m1 := [ [ 1, 0, 0, -7 ], 
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             [ 7, 1, 0, -7 ], 
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             [ 0, 7, 1, -7 ],
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             [ 0, 0, 7, -6 ] ];;
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gap> m2 := [ [ 0, 0, -13, 14 ], 
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             [ -1, 0, -13, 1 ], 
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             [ 13, -1, -13, 1 ], 
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             [ 0, 13, -14, 1 ] ];;
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gap> F := FieldByMatricesNC( [m1, m2] );
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<rational matrix field of unknown degree>
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gap> DegreeOverPrimeField(F);
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4
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gap> PrimitiveElement(F);
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[ [ -1, 1, 1, 0 ], [ -2, 0, 2, 1 ], [ -2, -1, 1, 2 ], [ -1, -1, 0, 1 ] ]
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gap> Basis(F);
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Basis( <rational matrix field of degree 4>, 
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[ [ [ 1, 0, 0, 0 ], [ 0, 1, 0, 0 ], [ 0, 0, 1, 0 ], [ 0, 0, 0, 1 ] ], 
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  [ [ 0, 1, 0, 0 ], [ -1, 1, 1, 0 ], [ -1, 0, 1, 1 ], [ -1, 0, 0, 1 ] ], 
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  [ [ 0, 0, 1, 0 ], [ -1, 0, 1, 1 ], [ -1, -1, 1, 1 ], [ 0, -1, 0, 1 ] ], 
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  [ [ 0, 0, 0, 1 ], [ -1, 0, 0, 1 ], [ 0, -1, 0, 1 ], [ 0, 0, -1, 1 ] ] ] )
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gap> MaximalOrderBasis(F);
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Basis( <rational matrix field of degree 4>, 
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[ [ [ 1, 0, 0, 0 ], [ 0, 1, 0, 0 ], [ 0, 0, 1, 0 ], [ 0, 0, 0, 1 ] ],
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  [ [ -1, 1, 1, 0 ], [ -2, 0, 2, 1 ], [ -2, -1, 1, 2 ], [ -1, -1, 0, 1 ] ],
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  [ [ -3, -2, 2, 3 ], [ -3, -5, 0, 5 ], [ 0, -5, -3, 3 ], [ 2, -2, -3, 0 ] ],
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  [ [ -1, -1, 0, 1 ], [ 0, -2, -1, 1 ], [ 1, -1, -2, 0 ], [ 1, 0, -1, -1 ] ]
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 ] )
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gap> U := UnitGroup(F);
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<matrix group with 2 generators>
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gap> u := GeneratorsOfGroup( U );;
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gap> nat := IsomorphismPcpGroup(U);;
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gap> H := Image(nat);
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Pcp-group with orders [ 10, 0 ]
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gap> ImageElm( nat, u[1] );
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g1
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gap> ImageElm( nat, u[2] );
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g2
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gap> ImageElm( nat, u[1]*u[2] );
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g1*g2
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gap> u[1] = PreImagesRepresentative(nat, GeneratorsOfGroup(H)[1] );
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true
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\endexample
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\Section{Number fields defined by a polynomial}
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\beginexample
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gap> g := UnivariatePolynomial( Rationals, [ 16, 64, -28, -4, 1 ] );
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x_1^4-4*x_1^3-28*x_1^2+64*x_1+16
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gap> F := FieldByPolynomialNC(g);
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<algebraic extension over the Rationals of degree 4>
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gap> PrimitiveElement(F);
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a
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gap> MaximalOrderBasis(F);
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Basis( <algebraic extension over the Rationals of degree 4>,
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[ !1, 1/2*a, 1/4*a^2, 1/56*a^3+1/14*a^2+1/14*a-2/7 ] )
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gap> U := UnitGroup(F);
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<group with 4 generators>
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gap> natU := IsomorphismPcpGroup(U);;
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gap> elms := List( [1..10], x-> Random(F) );;
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gap>  PcpPresentationOfMultiplicativeSubgroup( F, elms );
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Pcp-group with orders [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
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gap> isom := IsomorphismPcpGroup( F, elms );;
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gap> y := RandomGroupElement( elms );;
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gap> z := ImageElm( isom, y );;
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gap> y = PreImagesRepresentative( isom, z );
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true
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gap> FactorsPolynomialAlgExt( F, g );
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[ x_1+(-a), x_1+(a-2), x_1+(-1/7*a^3+3/7*a^2+31/7*a-40/7),
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  x_1+(1/7*a^3-3/7*a^2-31/7*a+26/7) ]
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\endexample
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