In this section we outline two example computations with the functions of the previous chapter. The first example uses number fields defined by matrices and the second example considers number fields defined by a polynomial.
gap> m1 := [ [ 1, 0, 0, -7 ], [ 7, 1, 0, -7 ], [ 0, 7, 1, -7 ], [ 0, 0, 7, -6 ] ];; gap> m2 := [ [ 0, 0, -13, 14 ], [ -1, 0, -13, 1 ], [ 13, -1, -13, 1 ], [ 0, 13, -14, 1 ] ];; gap> F := FieldByMatricesNC( [m1, m2] ); <field in characteristic 0> gap> DegreeOverPrimeField(F); 4 gap> PrimitiveElement(F); [ [ 1, 0, 0, -7 ], [ 7, 1, 0, -7 ], [ 0, 7, 1, -7 ], [ 0, 0, 7, -6 ] ] gap> Basis(F); Basis( <field in characteristic 0>, [ [ [ 1, 0, 0, 0 ], [ 0, 1, 0, 0 ], [ 0, 0, 1, 0 ], [ 0, 0, 0, 1 ] ], [ [ 0, 1, 0, 0 ], [ -1, 1, 1, 0 ], [ -1, 0, 1, 1 ], [ -1, 0, 0, 1 ] ], [ [ 0, 0, 1, 0 ], [ -1, 0, 1, 1 ], [ -1, -1, 1, 1 ], [ 0, -1, 0, 1 ] ], [ [ 0, 0, 0, 1 ], [ -1, 0, 0, 1 ], [ 0, -1, 0, 1 ], [ 0, 0, -1, 1 ] ] ] ) gap> MaximalOrderBasis(F); Basis( <field in characteristic 0>, [ [ [ 1, 0, 0, 0 ], [ 0, 1, 0, 0 ], [ 0, 0, 1, 0 ], [ 0, 0, 0, 1 ] ], [ [ 1, 0, 0, -1 ], [ 1, 1, 0, -1 ], [ 0, 1, 1, -1 ], [ 0, 0, 1, 0 ] ], [ [ 1, 0, -1, 0 ], [ 1, 1, -1, -1 ], [ 1, 1, 0, -1 ], [ 0, 1, 0, 0 ] ], [ [ 1, -1, 0, 0 ], [ 1, 0, -1, 0 ], [ 1, 0, 0, -1 ], [ 1, 0, 0, 0 ] ] ] ) gap> U := UnitGroup(F); <matrix group with 2 generators> gap> u := GeneratorsOfGroup( U );; gap> nat := IsomorphismPcpGroup(U); [ [ [ 0, 1, -1, 0 ], [ 0, 1, 0, -1 ], [ 0, 1, 0, 0 ], [ -1, 1, 0, 0 ] ], [ [ 1, 0, -1, 1 ], [ 0, 1, -1, 0 ], [ 1, 0, 0, 0 ], [ 0, 1, -1, 1 ] ] ] -> [ g1, g2 ] gap> H := Image(nat); Pcp-group with orders [ 10, 0 ] gap> ImageElm( nat, u[1]*u[2] ); g1*g2 gap> PreImagesRepresentative(nat, GeneratorsOfGroup(H)[1] ); [ [ 0, 1, -1, 0 ], [ 0, 1, 0, -1 ], [ 0, 1, 0, 0 ], [ -1, 1, 0, 0 ] ]
gap> x:=Indeterminate(Rationals); x_1 gap> g:= x^4-4*x^3-28*x^2+64*x+16; x_1^4-4*x_1^3-28*x_1^2+64*x_1+16 gap> F := FieldByPolynomialNC(g); <field in characteristic 0> gap> PrimitiveElement(F); (a) gap> MaximalOrderBasis(F); Basis( <field in characteristic 0>, [ !1, (1/2*a), (1/4*a^2), (5/7+1/14*a+1/14*a^2+1/56*a^3) ] ) gap> U := UnitGroup(F); [ !-1, (-3/7+6/7*a+3/28*a^2-1/28*a^3), (13/7+25/14*a+1/28*a^2-3/56*a^3), (36/7-9/7*a-2/7*a^2+3/56*a^3) ] <group with 4 generators> gap> natU := IsomorphismPcpGroup(U); [ !-1, (-3/7+6/7*a+3/28*a^2-1/28*a^3), (13/7+25/14*a+1/28*a^2-3/56*a^3), (36/7-9/7*a-2/7*a^2+3/56*a^3) ] -> [ g1, g2, g3, g4 ] gap> elms := List( [1..10], x-> Random(F) ); [ (4-1/2*a-1*a^2+3/2*a^3), (4/5-2/3*a+4/3*a^3), (1+a+2*a^2-1*a^3), (3/4+3*a+3*a^2), (-1-1/5*a^3), (-1/4*a-5/3*a^2), (1-1*a+1/2*a^2), (4-3/2*a+1/2*a^2), (-2/5+a-3/2*a^2), (-1*a+a^2+2*a^3) ] gap> PcpPresentationOfMultiplicativeSubgroup( F, elms ); Pcp-group with orders [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ] gap>isom := IsomorphismPcpGroup( F, elms ); [ (4-1/2*a-1*a^2+3/2*a^3), (4/5-2/3*a+4/3*a^3), (1+a+2*a^2-1*a^3), (3/4+3*a+3*a^2), (-1-1/5*a^3), (-1/4*a-5/3*a^2), (1-1*a+1/2*a^2), (4-3/2*a+1/2*a^2), (-2/5+a-3/2*a^2), (-1*a+a^2+2*a^3) ] [ (4-1/2*a-1*a^2+3/2*a^3), (4/5-2/3*a+4/3*a^3), (1+a+2*a^2-1*a^3), (3/4+3*a+3*a^2), (-1-1/5*a^3), (-1/4*a-5/3*a^2), (1-1*a+1/2*a^2), (4-3/2*a+1/2*a^2), (-2/5+a-3/2*a^2), (-1*a+a^2+2*a^3) ] -> [ g1, g2, g3, g4, g5, g6, g7, g8, g9, g10 ] gap> y := RandomGroupElement( elms ); (-475709724976707031371325/71806328788189775767952976 -379584641261299592239825/13055696143307231957809632*a -462249188570593771377595/287225315152759103071811904*a^2+ 2639763613873579813685/2901265809623829323957696*a^3) gap> ImageElm( isom, y ); g1^-1*g3^-2*g6^2*g8^-1*g9^-1 gap> z := last; g1^-1*g3^-2*g6^2*g8^-1*g9^-1 gap> PreImagesRepresentative( isom, z ); (-475709724976707031371325/71806328788189775767952976 -379584641261299592239825/13055696143307231957809632*a -462249188570593771377595/287225315152759103071811904*a^2+ 2639763613873579813685/2901265809623829323957696*a^3) gap> FactorsPolynomialKant( g, F ); [ x_1+(-40/7+31/7*a+3/7*a^2-1/7*a^3), x_1+(-2+a), x_1+(-1*a), x_1+(26/7-31/7*a-3/7*a^2+1/7*a^3) ]
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