Section 10, continued
gap> R1:= Integers mod 7; GF(7) gap> x:= X(R1, "x"); x gap> x^8-2*x; x^8+Z(7)^5*x
Exercise 10.1 gap> R1:= Integers mod 3; GF(3) gap> R2:= Integers mod 5; GF(5) gap> R3:= Integers mod 7; GF(7) gap> R4:= Integers mod 11; GF(11) gap> x:= X(R1, "x"); x gap> y:= X(R2, "y"); y gap> z:= X(R3, "z"); z gap> w:= X(R4, "w"); w gap> Factors(x^2-1); [ x+Z(3)^0, x-Z(3)^0 ] gap> Factors(y^4-1); [ y+Z(5)^0, y+Z(5), y-Z(5)^0, y+Z(5)^3 ] gap> Factors(z^6-1); [ z+Z(7)^0, z+Z(7), z+Z(7)^2, z-Z(7)^0, z+Z(7)^4, z+Z(7)^5 ] gap> Factors(w^10-1); [ w+Z(11)^0, w+Z(11), w+Z(11)^2, w+Z(11)^3, w+Z(11)^4, w-Z(11)^0, w+Z(11)^6, w+Z(11)^7, w+Z(11)^8, w+Z(11)^9 ]
Exercise 10.2 gap> IsIrreducible(x^3+2*x^2+1); true
Exercise 10.4 gap> R:= Rationals; Rationals gap> t:= X(R, "t"); t gap> IsIrreducible(t^3+2*t^2+1); true gap> IsIrreducible(x^4+x^3+x^2+1); true gap> IsIrreducible(t^4+t^3+t^2+1); true gap> IsIrreducible(y^4+y^2+2); true gap> IsIrreducible(t^4+t^2+2); true
Exercise 10.7 gap> Factors(t^6-1); [ t-1, t+1, t^2-t+1, t^2+t+1 ] gap> Factors(t^8-1); [ t-1, t+1, t^2+1, t^4+1 ] gap> Factors(t^12-1); [ t-1, t+1, t^2-t+1, t^2+1, t^2+t+1, t^4-t^2+1 ] gap> Factors(t^20-1); [ t-1, t+1, t^2+1, t^4-t^3+t^2-t+1, t^4+t^3+t^2+t+1, t^8-t^6+t^4-t^2+1 ] gap> Factors(t^30-1); [ t-1, t+1, t^2-t+1, t^2+t+1, t^4-t^3+t^2-t+1, t^4+t^3+t^2+t+1, t^8-t^7+t^5-t^4+t^3-t+1, t^8+t^7-t^5-t^4-t^3+t+1 ] gap> Factors(t^40-1); [ t-1, t+1, t^2+1, t^4-t^3+t^2-t+1, t^4+1, t^4+t^3+t^2+t+1, t^8-t^6+t^4-t^2+1, t^16-t^12+t^8-t^4+1 ] gap> Factors(t^50-1); [ t-1, t+1, t^4-t^3+t^2-t+1, t^4+t^3+t^2+t+1, t^20-t^15+t^10-t^5+1, t^20+t^15+t^10+t^5+1 ] gap> Factors(t^75-1); [ t-1, t^2+t+1, t^4+t^3+t^2+t+1, t^8-t^7+t^5-t^4+t^3-t+1, t^20+t^15+t^10+t^5+1, t^40-t^35+t^25-t^20+t^15-t^5+1 ] gap> Factors(t^105-1); [ t-1, t^2+t+1, t^4+t^3+t^2+t+1, t^6+t^5+t^4+t^3+t^2+t+1, t^8-t^7+t^5-t^4+t^3-t+1, t^12-t^11+t^9-t^8+t^6-t^4+t^3-t+1, t^24-t^23+t^19-t^18+t^17-t^16+t^14-t^13+t^12-t^11+t^10-t^8+t^7-t^6+t^5-t+1, t^48+t^47+t^46-t^43-t^42-2*t^41-t^40-t^39+t^36+t^35+t^34+t^33+t^32+t^31-t^28-t^26-t^24-t^22-t^20+t^17+t^16+t^15+t^14+t^13+t^12-t^\ 9-t^8-2*t^7-t^6-t^5+t^2+t+1 ] gap> Factors(t^106-1); [ t-1, t+1, t^52-t^51+t^50-t^49+t^48-t^47+t^46-t^45+t^44-t^43+t^42-t^41+t^40-t^39+t^38-t^37+t^36-t^35+t^34-t^33+t^32-t^31+t^30-t^29+t^28-t^27\ +t^26-t^25+t^24-t^23+t^22-t^21+t^20-t^19+t^18-t^17+t^16-t^15+t^14-t^13+t^12-t^11+t^10-t^9+t^8-t^7+t^6-t^5+t^4-t^3+t^2-t+1, t^52+t^51+t^50+t^49+t^48+t^47+t^46+t^45+t^44+t^43+t^42+t^41+t^40+t^39+t^38+t^37+t^36+t^35+t^34+t^33+t^32+t^31+t^30+t^29+t^28+t^27\ +t^26+t^25+t^24+t^23+t^22+t^21+t^20+t^19+t^18+t^17+t^16+t^15+t^14+t^13+t^12+t^11+t^10+t^9+t^8+t^7+t^6+t^5+t^4+t^3+t^2+t+1 ]
Section 11: 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);
[ <vector space of dimension 1 over GF(3)>, <vector space of dimension 1 over GF(3)>, <vector space of dimension 1 over GF(3)>,
<vector space of dimension 1 over GF(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> Size(Subspaces(V,1));
4
gap> HELP("Display");
gap> HELP("1");
Exercise 11.2 gap> V := GF(3)^3; ( GF(3)^3 ) gap> Elements(V); [ [ 0*Z(3), 0*Z(3), 0*Z(3) ], [ 0*Z(3), 0*Z(3), Z(3)^0 ], [ 0*Z(3), 0*Z(3), Z(3) ], [ 0*Z(3), Z(3)^0, 0*Z(3) ], [ 0*Z(3), Z(3)^0, Z(3)^0 ], [ 0*Z(3), Z(3)^0, Z(3) ], [ 0*Z(3), Z(3), 0*Z(3) ], [ 0*Z(3), Z(3), Z(3)^0 ], [ 0*Z(3), Z(3), Z(3) ], [ Z(3)^0, 0*Z(3), 0*Z(3) ], [ Z(3)^0, 0*Z(3), Z(3)^0 ], [ Z(3)^0, 0*Z(3), Z(3) ], [ Z(3)^0, Z(3)^0, 0*Z(3) ], [ Z(3)^0, Z(3)^0, Z(3)^0 ], [ Z(3)^0, Z(3)^0, Z(3) ], [ Z(3)^0, Z(3), 0*Z(3) ], [ Z(3)^0, Z(3), Z(3)^0 ], [ Z(3)^0, Z(3), Z(3) ], [ Z(3), 0*Z(3), 0*Z(3) ], [ Z(3), 0*Z(3), Z(3)^0 ], [ Z(3), 0*Z(3), Z(3) ], [ Z(3), Z(3)^0, 0*Z(3) ], [ Z(3), Z(3)^0, Z(3)^0 ], [ Z(3), Z(3)^0, Z(3) ], [ Z(3), Z(3), 0*Z(3) ], [ Z(3), Z(3), Z(3)^0 ], [ Z(3), Z(3), Z(3) ] ] gap> Display(Elements(V)); . . . . . 1 . . 2 . 1 . . 1 1 . 1 2 . 2 . . 2 1 . 2 2 1 . . 1 . 1 1 . 2 1 1 . 1 1 1 1 1 2 1 2 . 1 2 1 1 2 2 2 . . 2 . 1 2 . 2 2 1 . 2 1 1 2 1 2 2 2 . 2 2 1 2 2 2 gap> D := Subspaces(V,1); Subspaces( ( GF(3)^3 ), 1 ) gap> e := Elements(D); [ <vector space of dimension 1 over GF(3)>, <vector space of dimension 1 over GF(3)>, <vector space of dimension 1 over GF(3)>, <vector space of dimension 1 over GF(3)>, <vector space of dimension 1 over GF(3)>, <vector space of dimension 1 over GF(3)>, <vector space of dimension 1 over GF(3)>, <vector space of dimension 1 over GF(3)>, <vector space of dimension 1 over GF(3)>, <vector space of dimension 1 over GF(3)>, <vector space of dimension 1 over GF(3)>, <vector space of dimension 1 over GF(3)>, <vector space of dimension 1 over GF(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(V,1)); 13
Exercise 11.3 gap> V := GF(2)^3; ( GF(2)^3 ) gap> Size(Subspaces(V,1)); 7 gap> V := GF(5)^3; ( GF(5)^3 ) gap> Size(Subspaces(V,1)); 31 gap> V := GF(7)^3; ( GF(7)^3 ) gap> Size(Subspaces(V,1)); 57 gap> V := GF(11)^3; ( GF(11)^3 ) gap> Size(Subspaces(V,1)); 133 gap> 2^2+2+1; 7 gap> 3^2+3+1; 13 gap> 5^2+5+1; 31 gap> 7^2+7+1; 57 gap> 11^2+11+1; 133
Exercise 11.6 gap> V := GF(3)^3; ( GF(3)^3 ) gap> D := Subspaces(V,2); Subspaces( ( GF(3)^3 ), 2 ) gap> e := Elements(D); [ <vector space of dimension 2 over GF(3)>, <vector space of dimension 2 over GF(3)>, <vector space of dimension 2 over GF(3)>, <vector space of dimension 2 over GF(3)>, <vector space of dimension 2 over GF(3)>, <vector space of dimension 2 over GF(3)>, <vector space of dimension 2 over GF(3)>, <vector space of dimension 2 over GF(3)>, <vector space of dimension 2 over GF(3)>, <vector space of dimension 2 over GF(3)>, <vector space of dimension 2 over GF(3)>, <vector space of dimension 2 over GF(3)>, <vector space of dimension 2 over GF(3)> ] gap> Display(Elements(e)); [ VectorSpace( GF(3), [ [ 0*Z(3), Z(3)^0, 0*Z(3) ], [ 0*Z(3), 0*Z(3), Z(3)^0 ] ] ), VectorSpace( GF(3), [ [ Z(3)^0, 0*Z(3), 0*Z(3) ], [ 0*Z(3), 0*Z(3), Z(3)^0 ] ] ), VectorSpace( GF(3), [ [ Z(3)^0, Z(3)^0, 0*Z(3) ], [ 0*Z(3), 0*Z(3), Z(3)^0 ] ] ), VectorSpace( GF(3), [ [ Z(3)^0, Z(3), 0*Z(3) ], [ 0*Z(3), 0*Z(3), Z(3)^0 ] ] ), VectorSpace( GF(3), [ [ Z(3)^0, 0*Z(3), 0*Z(3) ], [ 0*Z(3), Z(3)^0, 0*Z(3) ] ] ), VectorSpace( GF(3), [ [ Z(3)^0, 0*Z(3), Z(3)^0 ], [ 0*Z(3), Z(3)^0, 0*Z(3) ] ] ), VectorSpace( GF(3), [ [ Z(3)^0, 0*Z(3), Z(3) ], [ 0*Z(3), Z(3)^0, 0*Z(3) ] ] ), VectorSpace( GF(3), [ [ Z(3)^0, 0*Z(3), 0*Z(3) ], [ 0*Z(3), Z(3)^0, Z(3)^0 ] ] ), VectorSpace( GF(3), [ [ Z(3)^0, 0*Z(3), Z(3)^0 ], [ 0*Z(3), Z(3)^0, Z(3)^0 ] ] ), VectorSpace( GF(3), [ [ Z(3)^0, 0*Z(3), Z(3) ], [ 0*Z(3), Z(3)^0, Z(3)^0 ] ] ), VectorSpace( GF(3), [ [ Z(3)^0, 0*Z(3), 0*Z(3) ], [ 0*Z(3), Z(3)^0, Z(3) ] ] ), VectorSpace( GF(3), [ [ Z(3)^0, 0*Z(3), Z(3)^0 ], [ 0*Z(3), Z(3)^0, Z(3) ] ] ), VectorSpace( GF(3), [ [ Z(3)^0, 0*Z(3), Z(3) ], [ 0*Z(3), Z(3)^0, Z(3) ] ] ) ] gap> Display(Elements(e[1])); . . . . . 1 . . 2 . 1 . . 1 1 . 1 2 . 2 . . 2 1 . 2 2 gap> Display(Elements(e[2])); . . . . . 1 . . 2 1 . . 1 . 1 1 . 2 2 . . 2 . 1 2 . 2 gap> Display(Elements(e[3])); . . . . . 1 . . 2 1 1 . 1 1 1 1 1 2 2 2 . 2 2 1 2 2 2 gap> Display(Elements(e[7])); . . . . 1 . . 2 . 1 . 2 1 1 2 1 2 2 2 . 1 2 1 1 2 2 1 gap> Display(Elements(e[11])); . . . . 1 2 . 2 1 1 . . 1 1 2 1 2 1 2 . . 2 1 2 2 2 1 gap> Display(Elements(e[13])); . . . . 1 2 . 2 1 1 . 2 1 1 1 1 2 . 2 . 1 2 1 . 2 2 2
Exercise 11.7 gap> V := GF(2)^3; ( GF(2)^3 ) gap> Size(Subspaces(V,2)); 7 gap> V := GF(5)^3; ( GF(5)^3 ) gap> Size(Subspaces(V,2)); 31 gap> V := GF(7)^3; ( GF(7)^3 ) gap> Size(Subspaces(V,2)); 57 gap> V := GF(11)^3; ( GF(11)^3 ) gap> Size(Subspaces(V,2)); 133
Section 12: Finite Fields
gap> Integers mod 11; GF(11) 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> F:= GF(3^4); GF(3^4) gap> Elements(F); [ 0*Z(3), Z(3)^0, Z(3), Z(3^2), Z(3^2)^2, Z(3^2)^3, Z(3^2)^5, Z(3^2)^6, Z(3^2)^7, Z(3^4), Z(3^4)^2, Z(3^4)^3, Z(3^4)^4, Z(3^4)^5, Z(3^4)^6, Z(3^4)^7, Z(3^4)^8, Z(3^4)^9, Z(3^4)^11, Z(3^4)^12, Z(3^4)^13, Z(3^4)^14, Z(3^4)^15, Z(3^4)^16, Z(3^4)^17, Z(3^4)^18, Z(3^4)^19, Z(3^4)^21, Z(3^4)^22, Z(3^4)^23, Z(3^4)^24, Z(3^4)^25, Z(3^4)^26, Z(3^4)^27, Z(3^4)^28, Z(3^4)^29, Z(3^4)^31, Z(3^4)^32, Z(3^4)^33, Z(3^4)^34, Z(3^4)^35, Z(3^4)^36, Z(3^4)^37, Z(3^4)^38, Z(3^4)^39, Z(3^4)^41, Z(3^4)^42, Z(3^4)^43, Z(3^4)^44, Z(3^4)^45, Z(3^4)^46, Z(3^4)^47, Z(3^4)^48, Z(3^4)^49, Z(3^4)^51, Z(3^4)^52, Z(3^4)^53, Z(3^4)^54, Z(3^4)^55, Z(3^4)^56, Z(3^4)^57, Z(3^4)^58, Z(3^4)^59, Z(3^4)^61, Z(3^4)^62, Z(3^4)^63, Z(3^4)^64, Z(3^4)^65, Z(3^4)^66, Z(3^4)^67, Z(3^4)^68, Z(3^4)^69, Z(3^4)^71, Z(3^4)^72, Z(3^4)^73, Z(3^4)^74, Z(3^4)^75, Z(3^4)^76, Z(3^4)^77, Z(3^4)^78, Z(3^4)^79 ] gap> DegreeFFE(Z(2^4)) > ; 4 gap> DegreeFFE(Z(2^4)^3); 4 gap> DegreeFFE(Z(2^4)^5); 2 gap> x:= X(GF(2), "x"); x gap> f:= x^4+x+1; x^4+x+Z(2)^0 gap> IsIrreducible(f); true gap> F:= AlgebraicExtension(GF(2), f); <field of size 16> gap> Factors(F,f); Error, no method found! For debugging hints type ?Recovery from NoMethodFound Error, no 2nd choice method found for `Factors' on 2 arguments called from <function>( <arguments> ) 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> Factors(f,F); Error, no method found! For debugging hints type ?Recovery from NoMethodFound Error, no 1st choice method found for `Factors' on 2 arguments called from <function>( <arguments> ) 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> P:= PolynomialRing(F); PolynomialRing(..., [ x_1 ]) gap> Factors(P,f); Error, no method found! For debugging hints type ?Recovery from NoMethodFound Error, no 2nd choice method found for `Factors' on 2 arguments called from <function>( <arguments> ) 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> Factors(f,P); Error, no method found! For debugging hints type ?Recovery from NoMethodFound Error, no 1st choice method found for `Factors' on 2 arguments called from <function>( <arguments> ) 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(F, "x"); x gap> f:= x^4+x+1; x^4+x+!Z(2)^0 gap> Factors(P,f); [ x+(a^2), x+(a), x+(Z(2)^0+a^2), x+(Z(2)^0+a) ] gap> quit;Back to Home Page
This PREP workshop is made possible by the NSF grant DUE: 0341481