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<html><head><title>[Alnuth] 3 An example application</title></head>
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<h1>3 An example application</h1><p>
<P>
<H3>Sections</H3>
<oL>
<li> <A HREF="CHAP003.htm#SECT001">Number fields defined by matrices</a>
<li> <A HREF="CHAP003.htm#SECT002">Number fields defined by a polynomial</a>
</ol><p>
<p>
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.
<p>
<p>
<h2><a name="SECT001">3.1 Number fields defined by matrices</a></h2>
<p><p>
<pre>
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] );
<rational matrix field of unknown degree>

gap> DegreeOverPrimeField(F);
4
gap> PrimitiveElement(F);
[ [ -1, 1, 1, 0 ], [ -2, 0, 2, 1 ], [ -2, -1, 1, 2 ], [ -1, -1, 0, 1 ] ]

gap> Basis(F);
Basis( <rational matrix field of degree 4>,
[ [ [ 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( <rational matrix field of degree 4>,
[ [ [ 1, 0, 0, 0 ], [ 0, 1, 0, 0 ], [ 0, 0, 1, 0 ], [ 0, 0, 0, 1 ] ],
  [ [ -1, 1, 1, 0 ], [ -2, 0, 2, 1 ], [ -2, -1, 1, 2 ], [ -1, -1, 0, 1 ] ],
  [ [ -3, -2, 2, 3 ], [ -3, -5, 0, 5 ], [ 0, -5, -3, 3 ], [ 2, -2, -3, 0 ] ],
  [ [ -1, -1, 0, 1 ], [ 0, -2, -1, 1 ], [ 1, -1, -2, 0 ], [ 1, 0, -1, -1 ] ]
] )

gap> U := UnitGroup(F);
<matrix group with 2 generators>

gap> u := GeneratorsOfGroup( U );;

gap> nat := IsomorphismPcpGroup(U);;
gap> H := Image(nat);
Pcp-group with orders [ 10, 0 ]
gap> ImageElm( nat, u[1] );
g1
gap> ImageElm( nat, u[2] );
g2
gap> ImageElm( nat, u[1]*u[2] );
g1*g2
gap> u[1] = PreImagesRepresentative(nat, GeneratorsOfGroup(H)[1] );
true
</pre>
<p>
<p>
<h2><a name="SECT002">3.2 Number fields defined by a polynomial</a></h2>
<p><p>
<pre>
gap> g := UnivariatePolynomial( Rationals, [ 16, 64, -28, -4, 1 ] );
x_1^4-4*x_1^3-28*x_1^2+64*x_1+16

gap> F := FieldByPolynomialNC(g);
<algebraic extension over the Rationals of degree 4>
gap> PrimitiveElement(F);
a
gap> MaximalOrderBasis(F);
Basis( <algebraic extension over the Rationals of degree 4>,
[ !1, 1/2*a, 1/4*a^2, 1/56*a^3+1/14*a^2+1/14*a-2/7 ] )

gap> U := UnitGroup(F);
<group with 4 generators>

gap> natU := IsomorphismPcpGroup(U);;
gap> elms := List( [1..10], x-> Random(F) );;

gap>  PcpPresentationOfMultiplicativeSubgroup( F, elms );
Pcp-group with orders [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]

gap> isom := IsomorphismPcpGroup( F, elms );;
gap> y := RandomGroupElement( elms );;
gap> z := ImageElm( isom, y );;
gap> y = PreImagesRepresentative( isom, z );
true

gap> FactorsPolynomialAlgExt( F, g );
[ x_1+(-a), x_1+(a-2), x_1+(-1/7*a^3+3/7*a^2+31/7*a-40/7),
  x_1+(1/7*a^3-3/7*a^2-31/7*a+26/7) ]
</pre>
<p>
<p>
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<P>
<address>Alnuth manual<br>April 2022
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