[Up] [Previous] [Next] [Index]

3 An example application

Sections

  1. Number fields defined by matrices
  2. Number fields defined by a polynomial

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.

3.1 Number fields defined by matrices

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 ] ]

3.2 Number fields defined by a polynomial

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) ]

[Up] [Previous] [Next] [Index]

ALNUTH manual
November 2006