1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
|
gap> START_TEST("Test of examples from the Alnuth manual");
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);
[ [ 0, -1, 1, 0 ], [ 0, -1, 0, 1 ], [ 0, -1, 0, 0 ], [ 1, -1, 0, 0 ] ]
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 ] ],
[ [ 0, -1, 1, 0 ], [ 0, -1, 0, 1 ], [ 0, -1, 0, 0 ], [ 1, -1, 0, 0 ] ],
[ [ 0, 0, 0, -1 ], [ 1, 0, 0, -1 ], [ 0, 1, 0, -1 ], [ 0, 0, 1, -1 ] ],
[ [ -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);;
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
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> Image(natU);
Pcp-group with orders [ 2, 0, 0, 0 ]
gap> elms := List( [1..10], x-> Random(F) );
[ -a^3+a^2-1/2*a-1, -3/4*a^3-a^2+1/3*a, -a^3-2*a^2-1, -2*a^3+a^2+a+1,
-3*a^2-1/3*a-4, 2/3*a^3+2*a^2-2/3*a-1, a^3-2*a^2+a, 2*a^3+2*a^2-2/3*a-2/3,
-4*a^2-4/3*a+3/2, -5/3*a+5/3 ]
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) ]
gap> STOP_TEST( "manual.tst", 100000);
|
