Section 12: Polynomial Rings

gap> f:=FreeGroup("a","b");

gap> AssignGeneratorVariables(f);
#I  Assigned the global variables [ a, b ]
gap> r:=[a^2,b^4,a^b/(a*b^2)];
[ a^2, b^4, b^-1*a*b^-1*a^-1 ]
gap> g:=f/r;

gap> AssignGeneratorVariables(g);
#I  Global variable `a' is already defined and will be overwritten
#I  Global variable `b' is already defined and will be overwritten
#I  Assigned the global variables [ a, b ]
gap> a^2;
a^2
gap> a^4;
a^4
gap> a^2=Identity of g;
Syntax error: ; expected
a^2=Identity of g;
              ^

gap> a^2=Identity(g);  
true
gap> StructureDescription(g);
"D8"
gap> r1:= Integers mod 7;
GF(7)
gap> p1:= PolynomialRing(r1);
GF(7)[x_1]
gap> x:= X(r1, "x");
x
gap> Factors(x^2-2);
[ x+Z(7), x+Z(7)^4 ]
gap> x:= X(Integers mod 8, "x");
x
gap> Factors(x^2-2);            
Error, no method found! For debugging hints type ?Recovery from NoMethodFound
Error, no 2nd choice method found for `Factors' on 2 arguments called from
Factors( DefaultRing( [ r ] ), r ) called from
(  ) called from read-eval-loop
Entering break read-eval-print loop ...
you can 'quit;' to quit to outer loop, or
you can 'return;' to continue
brk> quit;
gap> x:= X(Integers mod 5, "x");
x
gap> Factors(x^2-2);            
[ x^2+Z(5)^3 ]
gap> x:= X(Integers mod 11, "x");
x
gap> Factors(x^2-2);             
[ x^2+Z(11)^6 ]
gap> x:= X(Rationals, "x");      
x
gap> Factors(x^2-2);       
[ x^2-2 ]
gap> Factors(x^2-1);
[ x-1, x+1 ]
gap> Int(Z(11));
2
gap> Int(Z(11)^5);
10
gap> IsIrreducible(x^2-1);
false
gap> x:= X(Integer mod 3, "x");
Variable: 'Integer' must have a value

gap> x:= X(Integers mod 3, "x");
x
gap> Factors(x^2-1);            
[ x+Z(3)^0, x-Z(3)^0 ]
gap> x:= X(Integers mod 5, "x");
x
gap> Factors(x^2-1);            
[ x+Z(5)^0, x-Z(5)^0 ]
gap> Factors(x^4-1);
[ x+Z(5)^0, x+Z(5), x-Z(5)^0, x+Z(5)^3 ]
gap> x:= X(Integers mod 7, "x");
x
gap> Factors(x^6-1);            
[ x+Z(7)^0, x+Z(7), x+Z(7)^2, x-Z(7)^0, x+Z(7)^4, x+Z(7)^5 ]
gap> x:= X(Integers mod 11, "x");
x
gap> Factors(x^10-1);            
[ x+Z(11)^0, x+Z(11), x+Z(11)^2, x+Z(11)^3, x+Z(11)^4, x-Z(11)^0, x+Z(11)^6, 
  x+Z(11)^7, x+Z(11)^8, x+Z(11)^9 ]
gap> x:= X(Integers mod 3, "x"); 
x
gap> IsIrreducible(x^3 + 2*x^2 +1);
true
gap> IsIrreducible(x^4 + x^3+x^2 +1);
true
gap> x:= X(Rationals, "x");          
x
gap> IsIrreducible(x^4 + x^3+x^2 +1);
true
gap> IsIrreducible(x^3 + 2*x^2 +1);  
true
gap> x:= X(Rationals, "x");          
x
gap> Factors(x^6-1); 
[ x-1, x+1, x^2-x+1, x^2+x+1 ]
gap> Factors(x^8-1);
[ x-1, x+1, x^2+1, x^4+1 ]
gap> Factors(x^12-1);
[ x-1, x+1, x^2-x+1, x^2+1, x^2+x+1, x^4-x^2+1 ]
gap> Factors(x^20-1);
[ x-1, x+1, x^2+1, x^4-x^3+x^2-x+1, x^4+x^3+x^2+x+1, x^8-x^6+x^4-x^2+1 ]
gap> Factors(x^30-1);
[ x-1, x+1, x^2-x+1, x^2+x+1, x^4-x^3+x^2-x+1, x^4+x^3+x^2+x+1, 
  x^8-x^7+x^5-x^4+x^3-x+1, x^8+x^7-x^5-x^4-x^3+x+1 ]
gap> Factors(x^40-1);
[ x-1, x+1, x^2+1, x^4-x^3+x^2-x+1, x^4+1, x^4+x^3+x^2+x+1, x^8-x^6+x^4-x^2+1, 
  x^16-x^12+x^8-x^4+1 ]
gap> Factors(x^50-1);
[ x-1, x+1, x^4-x^3+x^2-x+1, x^4+x^3+x^2+x+1, x^20-x^15+x^10-x^5+1, 
  x^20+x^15+x^10+x^5+1 ]
gap> Factors(x^105-1);
[ x-1, x^2+x+1, x^4+x^3+x^2+x+1, x^6+x^5+x^4+x^3+x^2+x+1, x^8-x^7+x^5-x^4+x^3-x+1,
  x^12-x^11+x^9-x^8+x^6-x^4+x^3-x+1, 
  x^24-x^23+x^19-x^18+x^17-x^16+x^14-x^13+x^12-x^11+x^10-x^8+x^7-x^6+x^5-x+1, 
  x^48+x^47+x^46-x^43-x^42-2*x^41-x^40-x^39+x^36+x^35+x^34+x^33+x^32+x^31-x^28-x^2\
6-x^24-x^22-x^20+x^17+x^16+x^15+x^14+x^13+x^12-x^9-x^8-2*x^7-x^6-x^5+x^2+x+1 ]

Section 13: Vector Spaces

gap> V:=GF(3)^2;
( GF(3)^2 )
gap> Elements(V);
[ [ 0*Z(3), 0*Z(3) ], [ 0*Z(3), Z(3)^0 ], [ 0*Z(3), Z(3) ], [ Z(3)^0, 0*Z(3) ], 
  [ Z(3)^0, Z(3)^0 ], [ Z(3)^0, Z(3) ], [ Z(3), 0*Z(3) ], [ Z(3), Z(3)^0 ], 
  [ Z(3), Z(3) ] ]
gap> Display(Elements(V));
 . .
 . 1
 . 2
 1 .
 1 1
 1 2
 2 .
 2 1
 2 2
gap> d:=Subspaces(V,1);
Subspaces( ( GF(3)^2 ), 1 )
gap> e:=Elements(d);;
gap> Display(Elements(e));
[ VectorSpace( GF(3), [ [ 0*Z(3), Z(3)^0 ] ] ), 
  VectorSpace( GF(3), [ [ Z(3)^0, 0*Z(3) ] ] ), 
  VectorSpace( GF(3), [ [ Z(3)^0, Z(3)^0 ] ] ), 
  VectorSpace( GF(3), [ [ Z(3)^0, Z(3) ] ] ) ]
gap> e;
[ , , , 
   ]
gap> Display(Elements(e[1]));
 . .
 . 1
 . 2
gap> Display(Elements(e[2]));
 . .
 1 .
 2 .
gap> Display(Elements(e[3]));
 . .
 1 1
 2 2
gap> Display(Elements(e[4]));
 . .
 1 2
 2 1
gap> Size(Subspaces(V,1));
4
gap> V:=GF(3)^3;
( GF(3)^3 )
gap> d:=Subspaces(V,1);      
Subspaces( ( GF(3)^3 ), 1 )
gap> Size(Subspaces(V,1));
13
gap> e:=Elements(d);;        
gap> Display(Elements(e));   
[ VectorSpace( GF(3), [ [ 0*Z(3), 0*Z(3), Z(3)^0 ] ] ), 
  VectorSpace( GF(3), [ [ 0*Z(3), Z(3)^0, 0*Z(3) ] ] ), 
  VectorSpace( GF(3), [ [ 0*Z(3), Z(3)^0, Z(3)^0 ] ] ), 
  VectorSpace( GF(3), [ [ 0*Z(3), Z(3)^0, Z(3) ] ] ), 
  VectorSpace( GF(3), [ [ Z(3)^0, 0*Z(3), 0*Z(3) ] ] ), 
  VectorSpace( GF(3), [ [ Z(3)^0, 0*Z(3), Z(3)^0 ] ] ), 
  VectorSpace( GF(3), [ [ Z(3)^0, 0*Z(3), Z(3) ] ] ), 
  VectorSpace( GF(3), [ [ Z(3)^0, Z(3)^0, 0*Z(3) ] ] ), 
  VectorSpace( GF(3), [ [ Z(3)^0, Z(3)^0, Z(3)^0 ] ] ), 
  VectorSpace( GF(3), [ [ Z(3)^0, Z(3)^0, Z(3) ] ] ), 
  VectorSpace( GF(3), [ [ Z(3)^0, Z(3), 0*Z(3) ] ] ), 
  VectorSpace( GF(3), [ [ Z(3)^0, Z(3), Z(3)^0 ] ] ), 
  VectorSpace( GF(3), [ [ Z(3)^0, Z(3), Z(3) ] ] ) ]
gap> Display(Elements(e[1]));
 . . .
 . . 1
 . . 2
gap> Display(Elements(e[2]));
 . . .
 . 1 .
 . 2 .
gap> Display(Elements(e[3]));
 . . .
 . 1 1
 . 2 2
gap> Display(Elements(e[4]));
 . . .
 . 1 2
 . 2 1
gap> Display(Elements(e[5]));
 . . .
 1 . .
 2 . .
gap> Display(Elements(e[6]));
 . . .
 1 . 1
 2 . 2
gap> Display(Elements(e[7]));
 . . .
 1 . 2
 2 . 1
gap> Display(Elements(e[8]));
 . . .
 1 1 .
 2 2 .
gap> Display(Elements(e[9]));
 . . .
 1 1 1
 2 2 2
gap> Display(Elements(e[10]));
 . . .
 1 1 2
 2 2 1
gap> Display(Elements(e[11]));
 . . .
 1 2 .
 2 1 .
gap> Display(Elements(e[12]));
 . . .
 1 2 1
 2 1 2
gap> Display(Elements(e[13]));
 . . .
 1 2 2
 2 1 1
gap> Size(Subspaces(GF(2)^3,1));
7
gap> Size(Subspaces(GF(3)^3,1));
13
gap> Size(Subspaces(GF(5)^3,1));
31
gap> Size(Subspaces(GF(7)^3,1));
57
gap> Size(Subspaces(GF(11)^3,1));
133
gap> (11^3-1);
1330
gap> 7^3-1;
342
gap> (p^3-1)/(p-1);
Variable: 'p' must have a value

gap> V;
( GF(3)^3 )
gap> Size(Subspaces(V,2));
13
gap> d:=Subspaces(V,2);      
Subspaces( ( GF(3)^3 ), 2 )
gap> e:=Elements(d);
[ , , , 
  , , , 
  , , , 
  , , , 
   ]
gap> Display(Elements(e[1]));
 . . .
 . . 1
 . . 2
 . 1 .
 . 1 1
 . 1 2
 . 2 .
 . 2 1
 . 2 2
gap> Display(Elements(e[11]));
 . . .
 . 1 2
 . 2 1
 1 . .
 1 1 2
 1 2 1
 2 . .
 2 1 2
 2 2 1
gap> Size(Subspaces(GF(5)^3,2)); 
31
gap> Size(Subspaces(GF(7)^3,2));
57
gap> Size(Subspaces(GF(11)^3,2));
133

Section 14: Finite Fields

gap> F:= GF(2^4);
GF(2^4)
gap> Elements(F);
[ 0*Z(2), Z(2)^0, Z(2^2), Z(2^2)^2, Z(2^4), Z(2^4)^2, Z(2^4)^3, Z(2^4)^4, 
  Z(2^4)^6, Z(2^4)^7, Z(2^4)^8, Z(2^4)^9, Z(2^4)^11, Z(2^4)^12, Z(2^4)^13, 
  Z(2^4)^14 ]
gap> Elements(GF(2^2)^2);
[ [ 0*Z(2), 0*Z(2) ], [ 0*Z(2), Z(2)^0 ], [ 0*Z(2), Z(2^2) ], 
  [ 0*Z(2), Z(2^2)^2 ], [ Z(2)^0, 0*Z(2) ], [ Z(2)^0, Z(2)^0 ], 
  [ Z(2)^0, Z(2^2) ], [ Z(2)^0, Z(2^2)^2 ], [ Z(2^2), 0*Z(2) ], 
  [ Z(2^2), Z(2)^0 ], [ Z(2^2), Z(2^2) ], [ Z(2^2), Z(2^2)^2 ], 
  [ Z(2^2)^2, 0*Z(2) ], [ Z(2^2)^2, Z(2)^0 ], [ Z(2^2)^2, Z(2^2) ], 
  [ Z(2^2)^2, Z(2^2)^2 ] ]
gap> Z(2^4);
Z(2^4)
gap> Z(2^2)^2;
Z(2^2)^2
gap> Z(2^2)^2 = Z(2^4);
false
gap> Order(Z(2^4));
15
gap> Order(Z(2^2)^2);  
3
gap> Order(Z(3^2));
8
gap> Order(Z(3)^2);
1
gap> Order(Z(5^2));
24
gap> Order(Z(5)^2); 
2
gap> Elements( Ring([Z(2^4)]));
[ 0*Z(2), Z(2)^0, Z(2^2), Z(2^2)^2, Z(2^4), Z(2^4)^2, Z(2^4)^3, Z(2^4)^4, 
  Z(2^4)^6, Z(2^4)^7, Z(2^4)^8, Z(2^4)^9, Z(2^4)^11, Z(2^4)^12, Z(2^4)^13, 
  Z(2^4)^14 ]
gap> Elements( Ring([Z(2^2)^2]));
[ 0*Z(2), Z(2)^0, Z(2^2), Z(2^2)^2 ]
gap> Z(2^4)^10;
Z(2^2)^2
gap> Z(2^4)^15;
Z(2)^0
gap> Z(2^4)^5; 
Z(2^2)
gap> DegreeFFE(Z(2^4));
4
gap> DegreeFFE(Z(2^4)^3);
4
gap> DegreeFFE(Z(2^4)^2);
4
gap> x:= X(GF(2), "x");
x
gap> f:= x^4+x+1;
x^4+x+Z(2)^0
gap> Factors(f);
[ x^4+x+Z(2)^0 ]
gap> F:= AlgebraicExtension(GF(2), f);

gap> Elements(F);                     
Error, no method found! For debugging hints type ?Recovery from NoMethodFound
Error, no 3rd choice method found for `PrimitiveRoot' on 1 arguments called from
PrimitiveRoot( F ) called from
Basis( V ) called from
AsSSortedList( coll ) called from
(  ) called from read-eval-loop
Entering break read-eval-print loop ...
you can 'quit;' to quit to outer loop, or
you can 'return;' to continue
brk> quit;
gap> a:= RootOfDefiningPolynomial(F);
(a)
gap> a^2;
(a^2)
gap> a^5;
(a+a^2)