  
  [1X59 [33X[0;0YFinite Fields[133X[101X
  
  [33X[0;0YThis  chapter  describes  the  special functionality which exists in [5XGAP[105X for
  finite  fields  and  their elements. Of course the general functionality for
  fields (see Chapter [14X58[114X) also applies to finite fields.[133X
  
  [33X[0;0YIn  the  following,  the  term  [13Xfinite  field  element[113X is used to denote [5XGAP[105X
  objects  in  the  category  [2XIsFFE[102X  ([14X59.1-1[114X),  and [13Xfinite field[113X means a field
  consisting  of  such  elements.  Note  that in principle we must distinguish
  these  fields  from  (abstract) finite fields. For example, the image of the
  embedding  of  a finite field into a field of rational functions in the same
  characteristic is of course a finite field but its elements are not in [2XIsFFE[102X
  ([14X59.1-1[114X), and in fact [5XGAP[105X does currently not support such fields.[133X
  
  [33X[0;0YSpecial representations exist for row vectors and matrices over small finite
  fields (see sections [14X23.3[114X and [14X24.14[114X).[133X
  
  
  [1X59.1 [33X[0;0YFinite Field Elements[133X[101X
  
  [1X59.1-1 IsFFE[101X
  
  [33X[1;0Y[29X[2XIsFFE[102X( [3Xobj[103X ) [32X Category[133X
  [33X[1;0Y[29X[2XIsFFECollection[102X( [3Xobj[103X ) [32X Category[133X
  [33X[1;0Y[29X[2XIsFFECollColl[102X( [3Xobj[103X ) [32X Category[133X
  [33X[1;0Y[29X[2XIsFFECollCollColl[102X( [3Xobj[103X ) [32X Category[133X
  
  [33X[0;0YObjects  in  the  category  [2XIsFFE[102X  are  used to implement elements of finite
  fields. In this manual, the term [13Xfinite field element[113X always means an object
  in [2XIsFFE[102X. All finite field elements of the same characteristic form a family
  in  [5XGAP[105X  (see [14X13.1[114X).  Any  collection  of  finite field elements of the same
  characteristic  (see [2XIsCollection[102X  ([14X30.1-1[114X))  lies in [2XIsFFECollection[102X, and a
  collection  of  such  collections  (e.g., a matrix of finite field elements)
  lies in [2XIsFFECollColl[102X.[133X
  
  [1X59.1-2 Z[101X
  
  [33X[1;0Y[29X[2XZ[102X( [3Xp^d[103X ) [32X function[133X
  [33X[1;0Y[29X[2XZ[102X( [3Xp[103X, [3Xd[103X ) [32X function[133X
  
  [33X[0;0YFor  creating  elements  of  a finite field, the function [2XZ[102X can be used. The
  call  [10XZ([3Xp[103X[10X,[3Xd[103X[10X)[110X  (alternatively [10XZ([3Xp[103X[10X^[3Xd[103X[10X)[110X) returns the designated generator of the
  multiplicative  group  of  the  finite  field with [3Xp^d[103X elements. [3Xp[103X must be a
  prime integer.[133X
  
  [33X[0;0Y[5XGAP[105X can represent elements of all finite fields [10XGF([3Xp^d[103X[10X)[110X such that either (1)
  [3Xp^d[103X  [22X<=  65536[122X (in which case an extremely efficient internal representation
  is  used);  (2) d = 1, (in which case, for large [3Xp[103X, the field is represented
  using  the machinery of residue class rings (see section [14X14.5[114X) or (3) if the
  Conway  polynomial  of  degree [3Xd[103X over the field with [3Xp[103X elements is known, or
  can be computed (see [2XConwayPolynomial[102X ([14X59.5-1[114X)).[133X
  
  [33X[0;0YIf  you  attempt  to construct an element of [10XGF([3Xp^d[103X[10X)[110X for which [3Xd[103X [22X> 1[122X and the
  relevant  Conway  polynomial  is not known, and not necessarily easy to find
  (see [2XIsCheapConwayPolynomial[102X ([14X59.5-2[114X)), then [5XGAP[105X will stop with an error and
  enter  the  break loop. If you leave this break loop by entering [10Xreturn;[110X [5XGAP[105X
  will  attempt  to  compute the Conway polynomial, which may take a very long
  time.[133X
  
  [33X[0;0YThe  root  returned  by  [2XZ[102X is a generator of the multiplicative group of the
  finite field with [3Xp^d[103X elements, which is cyclic. The order of the element is
  of  course  [3Xp^d[103X  [22X-1[122X. The [3Xp^d[103X [22X-1[122X different powers of the root are exactly the
  nonzero elements of the finite field.[133X
  
  [33X[0;0YThus  all  nonzero  elements  of  the  finite field with [3Xp^d[103X elements can be
  entered as [10XZ([3Xp^d[103X[10X)^[110X[22Xi[122X. Note that this is also the form that [5XGAP[105X uses to output
  those  elements  when  they  are  stored  in the internal representation. In
  larger  fields,  it is more convenient to enter and print elements as linear
  combinations of powers of the primitive element, see section [14X59.6[114X.[133X
  
  [33X[0;0YThe additive neutral element is [10X0 * Z([3Xp[103X[10X)[110X. It is different from the integer [10X0[110X
  in  subtle  ways.  First [10XIsInt( 0 * Z([3Xp[103X[10X) )[110X (see [2XIsInt[102X ([14X14.2-1[114X)) is [9Xfalse[109X and
  [10XIsFFE( 0 * Z([3Xp[103X[10X) )[110X (see [2XIsFFE[102X ([14X59.1-1[114X)) is [9Xtrue[109X, whereas it is just the other
  way around for the integer [10X0[110X.[133X
  
  [33X[0;0YThe  multiplicative  neutral  element  is  [10XZ([3Xp[103X[10X)^0[110X.  It is different from the
  integer  [10X1[110X  in  subtle  ways.  First [10XIsInt( Z([3Xp[103X[10X)^0 )[110X (see [2XIsInt[102X ([14X14.2-1[114X)) is
  [9Xfalse[109X  and  [10XIsFFE( Z([3Xp[103X[10X)^0 )[110X (see [2XIsFFE[102X ([14X59.1-1[114X)) is [9Xtrue[109X, whereas it is just
  the other way around for the integer [10X1[110X. Also [10X1+1[110X is [10X2[110X, whereas, e.g., [10XZ(2)^0
  + Z(2)^0[110X is [10X0 * Z(2)[110X.[133X
  
  [33X[0;0YThe various roots returned by [2XZ[102X for finite fields of the same characteristic
  are compatible in the following sense. If the field [10XGF([3Xp[103X[10X,[110X[22Xn[122X[10X)[110X is a subfield of
  the field [10XGF([3Xp[103X[10X,[110X[22Xm[122X[10X)[110X, i.e., [22Xn[122X divides [22Xm[122X, then [10XZ[110X[22X([3Xp[103X^n) =[122X[10XZ[110X[22X([3Xp[103X^m)^{([3Xp[103X^m-1)/([3Xp[103X^n-1)}[122X.
  Note that this is the simplest relation that may hold between a generator of
  [10XGF([3Xp[103X[10X,[110X[22Xn[122X[10X)[110X and [10XGF([3Xp[103X[10X,[110X[22Xm[122X[10X)[110X, since [10XZ[110X[22X([3Xp[103X^n)[122X is an element of order [22X[3Xp[103X^m-1[122X and [10XZ[110X[22X([3Xp[103X^m)[122X is
  an element of order [22X[3Xp[103X^n-1[122X. This is achieved by choosing [10XZ([3Xp[103X[10X)[110X as the smallest
  primitive  root  modulo [3Xp[103X and [10XZ([110X[3Xp^n[103X[10X)[110X as a root of the [22Xn[122X-th [13XConway polynomial[113X
  (see [2XConwayPolynomial[102X  ([14X59.5-1[114X)) of characteristic [3Xp[103X. Those polynomials were
  defined by J. H. Conway, and many of them were computed by R. A. Parker.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xa:= Z( 32 );[127X[104X
    [4X[28XZ(2^5)[128X[104X
    [4X[25Xgap>[125X [27Xa+a;[127X[104X
    [4X[28X0*Z(2)[128X[104X
    [4X[25Xgap>[125X [27Xa*a;[127X[104X
    [4X[28XZ(2^5)^2[128X[104X
    [4X[25Xgap>[125X [27Xb := Z(3,12);[127X[104X
    [4X[28Xz[128X[104X
    [4X[25Xgap>[125X [27Xb*b;[127X[104X
    [4X[28Xz2[128X[104X
    [4X[25Xgap>[125X [27Xb+b;[127X[104X
    [4X[28X2z[128X[104X
    [4X[25Xgap>[125X [27XPrint(b^100,"\n");[127X[104X
    [4X[28XZ(3)^0+Z(3,12)^5+Z(3,12)^6+2*Z(3,12)^8+Z(3,12)^10+Z(3,12)^11[128X[104X
  [4X[32X[104X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XZ(11,40);[127X[104X
    [4X[28XError, Conway Polynomial 11^40 will need to computed and might be slow[128X[104X
    [4X[28Xreturn to continue called from[128X[104X
    [4X[28XFFECONWAY.ZNC( p, d ) called from[128X[104X
    [4X[28X<function>( <arguments> ) called from read-eval-loop[128X[104X
    [4X[28XEntering break read-eval-print loop ...[128X[104X
    [4X[28Xyou can 'quit;' to quit to outer loop, or[128X[104X
    [4X[28Xyou can 'return;' to continue[128X[104X
    [4X[26Xbrk>[126X[104X
  [4X[32X[104X
  
  [1X59.1-3 IsLexOrderedFFE[101X
  
  [33X[1;0Y[29X[2XIsLexOrderedFFE[102X( [3Xffe[103X ) [32X Category[133X
  [33X[1;0Y[29X[2XIsLogOrderedFFE[102X( [3Xffe[103X ) [32X Category[133X
  
  [33X[0;0YElements  of  finite fields can be compared using the operators [10X=[110X and [10X<[110X. The
  call [10X[3Xa[103X[10X = [3Xb[103X[10X[110X returns [9Xtrue[109X if and only if the finite field elements [3Xa[103X and [3Xb[103X are
  equal.  Furthermore  [10X[3Xa[103X[10X  <  [3Xb[103X[10X[110X  tests  whether  [3Xa[103X is smaller than [3Xb[103X. The exact
  behaviour  of  this  comparison depends on which of two categories the field
  elements belong to:[133X
  
  [33X[0;0YFinite  field  elements  are  ordered  in  [5XGAP[105X  (by  [2X\<[102X  ([14X31.11-1[114X)) first by
  characteristic  and  then  by  their  degree (i.e. the sizes of the smallest
  fields  containing them). Amongst irreducible elements of a given field, the
  ordering  depends  on  which  of  these categories the elements of the field
  belong  to  (all  irreducible elements of a given field should belong to the
  same one)[133X
  
  [33X[0;0YElements   in   [2XIsLexOrderedFFE[102X   are  ordered  lexicographically  by  their
  coefficients with respect to the canonical basis of the field.[133X
  
  [33X[0;0YElements   in  [2XIsLogOrderedFFE[102X  are  ordered  according  to  their  discrete
  logarithms  with  respect  to the [2XPrimitiveElement[102X ([14X58.2-3[114X) attribute of the
  field.  For  the comparison of finite field elements with other [5XGAP[105X objects,
  see [14X4.13[114X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XZ( 16 )^10 = Z( 4 )^2;  # illustrates embedding of GF(4) in GF(16)[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27X0 < 0*Z(101);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XZ(256) > Z(101);[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XZ(2,20) < Z(2,20)^2; # this illustrates the lexicographic ordering[127X[104X
    [4X[28Xfalse[128X[104X
  [4X[32X[104X
  
  
  [1X59.2 [33X[0;0YOperations for Finite Field Elements[133X[101X
  
  [33X[0;0YSince  finite  field  elements  are  scalars,  the operations [2XCharacteristic[102X
  ([14X31.10-1[114X), [2XOne[102X ([14X31.10-2[114X), [2XZero[102X ([14X31.10-3[114X), [2XInverse[102X ([14X31.10-8[114X), [2XAdditiveInverse[102X
  ([14X31.10-9[114X),  [2XOrder[102X ([14X31.10-10[114X) can be applied to them (see [14X31.10[114X). Contrary to
  the  situation  with other scalars, [2XOrder[102X ([14X31.10-10[114X) is defined also for the
  zero element in a finite field, with value [10X0[110X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XCharacteristic( Z( 16 )^10 );  Characteristic( Z( 9 )^2 );[127X[104X
    [4X[28X2[128X[104X
    [4X[28X3[128X[104X
    [4X[25Xgap>[125X [27XCharacteristic( [ Z(4), Z(8) ] );[127X[104X
    [4X[28X2[128X[104X
    [4X[25Xgap>[125X [27XOne( Z(9) );  One( 0*Z(4) );[127X[104X
    [4X[28XZ(3)^0[128X[104X
    [4X[28XZ(2)^0[128X[104X
    [4X[25Xgap>[125X [27XInverse( Z(9) );  AdditiveInverse( Z(9) );[127X[104X
    [4X[28XZ(3^2)^7[128X[104X
    [4X[28XZ(3^2)^5[128X[104X
    [4X[25Xgap>[125X [27XOrder( Z(9)^7 );[127X[104X
    [4X[28X8[128X[104X
  [4X[32X[104X
  
  [1X59.2-1 DegreeFFE[101X
  
  [33X[1;0Y[29X[2XDegreeFFE[102X( [3Xz[103X ) [32X attribute[133X
  [33X[1;0Y[29X[2XDegreeFFE[102X( [3Xvec[103X ) [32X method[133X
  [33X[1;0Y[29X[2XDegreeFFE[102X( [3Xmat[103X ) [32X method[133X
  
  [33X[0;0Y[2XDegreeFFE[102X  returns  the degree of the smallest finite field [3XF[103X containing the
  element  [3Xz[103X,  respectively  all  elements of the row vector [3Xvec[103X over a finite
  field (see [14X23[114X), or the matrix [3Xmat[103X over a finite field (see [14X24[114X).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XDegreeFFE( Z( 16 )^10 );[127X[104X
    [4X[28X2[128X[104X
    [4X[25Xgap>[125X [27XDegreeFFE( Z( 16 )^11 );[127X[104X
    [4X[28X4[128X[104X
    [4X[25Xgap>[125X [27XDegreeFFE( [ Z(2^13), Z(2^10) ] );[127X[104X
    [4X[28X130[128X[104X
  [4X[32X[104X
  
  [1X59.2-2 LogFFE[101X
  
  [33X[1;0Y[29X[2XLogFFE[102X( [3Xz[103X, [3Xr[103X ) [32X operation[133X
  
  [33X[0;0Y[2XLogFFE[102X  returns  the  discrete  logarithm of the element [3Xz[103X in a finite field
  with  respect  to  the  root  [3Xr[103X. An error is signalled if [3Xz[103X is zero. [9Xfail[109X is
  returned if [3Xz[103X is not a power of [3Xr[103X.[133X
  
  [33X[0;0YThe  [13Xdiscrete  logarithm[113X  of the element [3Xz[103X with respect to the root [3Xr[103X is the
  smallest nonnegative integer [22Xi[122X such that [22X[3Xr[103X^i = [3Xz[103X[122X holds.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XLogFFE( Z(409)^116, Z(409) );  LogFFE( Z(409)^116, Z(409)^2 );[127X[104X
    [4X[28X116[128X[104X
    [4X[28X58[128X[104X
  [4X[32X[104X
  
  [1X59.2-3 IntFFE[101X
  
  [33X[1;0Y[29X[2XIntFFE[102X( [3Xz[103X ) [32X attribute[133X
  [33X[1;0Y[29X[2XInt[102X( [3Xz[103X ) [32X method[133X
  
  [33X[0;0Y[2XIntFFE[102X returns the integer corresponding to the element [3Xz[103X, which must lie in
  a  finite  prime  field.  That  is,  [2XIntFFE[102X returns the smallest nonnegative
  integer [22Xi[122X such that [22Xi[122X[10X * One( [110X[3Xz[103X[10X ) = [110X[3Xz[103X.[133X
  
  [33X[0;0YThe   correspondence   between   elements  from  a  finite  prime  field  of
  characteristic  [22Xp[122X  (for  [22Xp  <  2^16[122X)  and  the integers between [22X0[122X and [22Xp-1[122X is
  defined by choosing [10XZ([110X[22Xp[122X[10X)[110X the element corresponding to the smallest primitive
  root mod [22Xp[122X (see [2XPrimitiveRootMod[102X ([14X15.3-3[114X)).[133X
  
  [33X[0;0Y[2XIntFFE[102X is installed as a method for the operation [2XInt[102X ([14X14.2-3[114X) with argument
  a finite field element.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XIntFFE( Z(13) );  PrimitiveRootMod( 13 );[127X[104X
    [4X[28X2[128X[104X
    [4X[28X2[128X[104X
    [4X[25Xgap>[125X [27XIntFFE( Z(409) );[127X[104X
    [4X[28X21[128X[104X
    [4X[25Xgap>[125X [27XIntFFE( Z(409)^116 );  21^116 mod 409;[127X[104X
    [4X[28X311[128X[104X
    [4X[28X311[128X[104X
  [4X[32X[104X
  
  [33X[0;0YSee also [2XIntFFESymm[102X ([14X59.2-4[114X).[133X
  
  [1X59.2-4 IntFFESymm[101X
  
  [33X[1;0Y[29X[2XIntFFESymm[102X( [3Xz[103X ) [32X attribute[133X
  [33X[1;0Y[29X[2XIntFFESymm[102X( [3Xvec[103X ) [32X attribute[133X
  
  [33X[0;0YFor  a  finite  prime  field element [3Xz[103X, [2XIntFFESymm[102X returns the corresponding
  integer  of smallest absolute value. That is, [2XIntFFESymm[102X returns the integer
  [22Xi[122X of smallest absolute value such that [22Xi[122X[10X * One( [110X[3Xz[103X[10X ) = [110X[3Xz[103X holds.[133X
  
  [33X[0;0YFor  a  vector  [3Xvec[103X  of  FFEs,  the operation returns the result of applying
  [2XIntFFESymm[102X to every entry of the vector.[133X
  
  [33X[0;0YThe   correspondence   between   elements  from  a  finite  prime  field  of
  characteristic  [22Xp[122X  (for  [22Xp  < 2^16[122X) and the integers between [22X-p/2[122X and [22Xp/2[122X is
  defined  by choosing [10XZ([110X[22Xp[122X[10X)[110X the element corresponding to the smallest positive
  primitive root mod [22Xp[122X (see [2XPrimitiveRootMod[102X ([14X15.3-3[114X)) and reducing results to
  the [22X-p/2 .. p/2[122X range.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XIntFFE(Z(13)^2);IntFFE(Z(13)^3);[127X[104X
    [4X[28X4[128X[104X
    [4X[28X8[128X[104X
    [4X[25Xgap>[125X [27XIntFFESymm(Z(13)^2);IntFFESymm(Z(13)^3);[127X[104X
    [4X[28X4[128X[104X
    [4X[28X-5[128X[104X
  [4X[32X[104X
  
  [33X[0;0YSee also [2XIntFFE[102X ([14X59.2-3[114X)[133X
  
  [1X59.2-5 IntVecFFE[101X
  
  [33X[1;0Y[29X[2XIntVecFFE[102X( [3Xvecffe[103X ) [32X operation[133X
  
  [33X[0;0Yis  the  list of integers corresponding to the vector [3Xvecffe[103X of finite field
  elements in a prime field (see [2XIntFFE[102X ([14X59.2-3[114X)).[133X
  
  [1X59.2-6 AsInternalFFE[101X
  
  [33X[1;0Y[29X[2XAsInternalFFE[102X( [3Xffe[103X ) [32X attribute[133X
  
  [33X[0;0Yreturn an internal FFE equal to [3Xffe[103X if one exists, otherwise [10Xfail[110X[133X
  
  [1X59.2-7 RootFFE[101X
  
  [33X[1;0Y[29X[2XRootFFE[102X( [3XF[103X, [3Xz[103X, [3Xk[103X ) [32X operation[133X
  
  [33X[0;0Y[2XRootFFE[102X returns a finite field element [3Xr[103X from [3XF[103X whose [3Xk[103X-th power is [3Xz[103X. If no
  such element exists then [9Xfail[109X is returned.[133X
  
  
  [1X59.3 [33X[0;0YCreating Finite Fields[133X[101X
  
  [1X59.3-1 DefaultField[101X
  
  [33X[1;0Y[29X[2XDefaultField[102X( [3Xlist[103X ) [32X function[133X
  [33X[1;0Y[29X[2XDefaultRing[102X( [3Xlist[103X ) [32X function[133X
  
  [33X[0;0Y[2XDefaultField[102X and [2XDefaultRing[102X for finite field elements are defined to return
  the [13Xsmallest[113X field containing the given elements.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XDefaultField( [ Z(4), Z(4)^2 ] );  DefaultField( [ Z(4), Z(8) ] );[127X[104X
    [4X[28XGF(2^2)[128X[104X
    [4X[28XGF(2^6)[128X[104X
  [4X[32X[104X
  
  [1X59.3-2 GaloisField[101X
  
  [33X[1;0Y[29X[2XGaloisField[102X( [3Xp^d[103X ) [32X function[133X
  [33X[1;0Y[29X[2XGF[102X( [3Xp^d[103X ) [32X function[133X
  [33X[1;0Y[29X[2XGaloisField[102X( [3Xp[103X, [3Xd[103X ) [32X function[133X
  [33X[1;0Y[29X[2XGF[102X( [3Xp[103X, [3Xd[103X ) [32X function[133X
  [33X[1;0Y[29X[2XGaloisField[102X( [3Xsubfield[103X, [3Xd[103X ) [32X function[133X
  [33X[1;0Y[29X[2XGF[102X( [3Xsubfield[103X, [3Xd[103X ) [32X function[133X
  [33X[1;0Y[29X[2XGaloisField[102X( [3Xp[103X, [3Xpol[103X ) [32X function[133X
  [33X[1;0Y[29X[2XGF[102X( [3Xp[103X, [3Xpol[103X ) [32X function[133X
  [33X[1;0Y[29X[2XGaloisField[102X( [3Xsubfield[103X, [3Xpol[103X ) [32X function[133X
  [33X[1;0Y[29X[2XGF[102X( [3Xsubfield[103X, [3Xpol[103X ) [32X function[133X
  
  [33X[0;0Y[2XGaloisField[102X  returns  a  finite  field.  It  takes  two  arguments. The form
  [10XGaloisField(  [3Xp[103X[10X,  [3Xd[103X[10X  )[110X,  where  [3Xp[103X,  [3Xd[103X  are  integers,  can  also be given as
  [10XGaloisField( [3Xp[103X[10X^[3Xd[103X[10X )[110X. [2XGF[102X is an abbreviation for [2XGaloisField[102X.[133X
  
  [33X[0;0YThe  first  argument specifies the subfield [22XS[122X over which the new field is to
  be  taken.  It can be a prime integer or a finite field. If it is a prime [3Xp[103X,
  the subfield is the prime field of this characteristic.[133X
  
  [33X[0;0YThe  second  argument  specifies  the  extension. It can be an integer or an
  irreducible  polynomial  over  the  field  [22XS[122X. If it is an integer [3Xd[103X, the new
  field   is  constructed  as  the  polynomial  extension  w.r.t.  the  Conway
  polynomial  (see [2XConwayPolynomial[102X  ([14X59.5-1[114X)) of degree [3Xd[103X over [22XS[122X. If it is an
  irreducible  polynomial  [3Xpol[103X  over  [22XS[122X,  the  new  field  is  constructed  as
  polynomial  extension  of  [22XS[122X  with  this  polynomial;  in  this case, [3Xpol[103X is
  accessible  as  the  value of [2XDefiningPolynomial[102X ([14X58.2-7[114X) for the new field,
  and  a  root  of  [3Xpol[103X  in  the  new  field  is  accessible  as  the value of
  [2XRootOfDefiningPolynomial[102X ([14X58.2-8[114X).[133X
  
  [33X[0;0YNote  that  the  subfield over which a field was constructed determines over
  which  field  the Galois group, conjugates, norm, trace, minimal polynomial,
  and  trace  polynomial  are  computed  (see [2XGaloisGroup[102X ([14X58.3-1[114X), [2XConjugates[102X
  ([14X58.3-6[114X),   [2XNorm[102X   ([14X58.3-4[114X),  [2XTrace[102X  ([14X58.3-5[114X),  [2XMinimalPolynomial[102X  ([14X58.3-2[114X),
  [2XTracePolynomial[102X ([14X58.3-3[114X)).[133X
  
  [33X[0;0YThe field is regarded as a vector space (see [14X61[114X) over the given subfield, so
  this determines the dimension and the canonical basis of the field.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xf1:= GF( 2^4 );[127X[104X
    [4X[28XGF(2^4)[128X[104X
    [4X[25Xgap>[125X [27XSize( GaloisGroup ( f1 ) );[127X[104X
    [4X[28X4[128X[104X
    [4X[25Xgap>[125X [27XBasisVectors( Basis( f1 ) );[127X[104X
    [4X[28X[ Z(2)^0, Z(2^4), Z(2^4)^2, Z(2^4)^3 ][128X[104X
    [4X[25Xgap>[125X [27Xf2:= GF( GF(4), 2 );[127X[104X
    [4X[28XAsField( GF(2^2), GF(2^4) )[128X[104X
    [4X[25Xgap>[125X [27XSize( GaloisGroup( f2 ) );[127X[104X
    [4X[28X2[128X[104X
    [4X[25Xgap>[125X [27XBasisVectors( Basis( f2 ) );[127X[104X
    [4X[28X[ Z(2)^0, Z(2^4) ][128X[104X
  [4X[32X[104X
  
  [1X59.3-3 PrimitiveRoot[101X
  
  [33X[1;0Y[29X[2XPrimitiveRoot[102X( [3XF[103X ) [32X attribute[133X
  
  [33X[0;0YA  [13Xprimitive  root[113X  of  a  finite field is a generator of its multiplicative
  group.  A primitive root is always a primitive element (see [2XPrimitiveElement[102X
  ([14X58.2-3[114X)), the converse is in general not true.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xf:= GF( 3^5 );[127X[104X
    [4X[28XGF(3^5)[128X[104X
    [4X[25Xgap>[125X [27XPrimitiveRoot( f );[127X[104X
    [4X[28XZ(3^5)[128X[104X
  [4X[32X[104X
  
  
  [1X59.4 [33X[0;0YFrobenius Automorphisms[133X[101X
  
  [1X59.4-1 FrobeniusAutomorphism[101X
  
  [33X[1;0Y[29X[2XFrobeniusAutomorphism[102X( [3XF[103X ) [32X attribute[133X
  
  [33X[0;0Yreturns  the  Frobenius  automorphism  of  the  finite  field  [3XF[103X  as a field
  homomorphism (see [14X32.12[114X).[133X
  
  [33X[0;0YThe  [13XFrobenius automorphism[113X [22Xf[122X of a finite field [22XF[122X of characteristic [22Xp[122X is the
  function  that  takes  each  element  [22Xz[122X  of  [22XF[122X to its [22Xp[122X-th power. Each field
  automorphism  of  [22XF[122X  is  a  power of [22Xf[122X. Thus [22Xf[122X is a generator for the Galois
  group  of  [22XF[122X relative to the prime field of [22XF[122X, and an appropriate power of [22Xf[122X
  is  a  generator  of  the Galois group of [22XF[122X over a subfield (see [2XGaloisGroup[102X
  ([14X58.3-1[114X)).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xf := GF(16);[127X[104X
    [4X[28XGF(2^4)[128X[104X
    [4X[25Xgap>[125X [27Xx := FrobeniusAutomorphism( f );[127X[104X
    [4X[28XFrobeniusAutomorphism( GF(2^4) )[128X[104X
    [4X[25Xgap>[125X [27XZ(16) ^ x;[127X[104X
    [4X[28XZ(2^4)^2[128X[104X
    [4X[25Xgap>[125X [27Xx^2;[127X[104X
    [4X[28XFrobeniusAutomorphism( GF(2^4) )^2[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  image  of  an  element  [22Xz[122X  under the [22Xi[122X-th power of [22Xf[122X is computed as the
  [22Xp^i[122X-th  power of [22Xz[122X. The product of the [22Xi[122X-th power and the [22Xj[122X-th power of [22Xf[122X is
  the  [22Xk[122X-th power of [22Xf[122X, where [22Xk[122X is [22Xi j mod[122X [10XSize([3XF[103X[10X)[110X[22X-1[122X. The zeroth power of [22Xf[122X is
  [10XIdentityMapping( [3XF[103X[10X )[110X.[133X
  
  
  [1X59.5 [33X[0;0YConway Polynomials[133X[101X
  
  [1X59.5-1 ConwayPolynomial[101X
  
  [33X[1;0Y[29X[2XConwayPolynomial[102X( [3Xp[103X, [3Xn[103X ) [32X function[133X
  
  [33X[0;0Yis  the Conway polynomial of the finite field [22XGF(p^n)[122X as polynomial over the
  prime field in characteristic [3Xp[103X.[133X
  
  [33X[0;0YThe  [13XConway  polynomial[113X  [22XΦ_{n,p}[122X  of  [22XGF(p^n)[122X  is  defined  by the following
  properties.[133X
  
  [33X[0;0YFirst  define an ordering of polynomials of degree [22Xn[122X over [22XGF(p)[122X, as follows.
  [22Xf  =  ∑_{i  = 0}^n (-1)^i f_i x^i[122X is smaller than [22Xg = ∑_{i = 0}^n (-1)^i g_i
  x^i[122X  if  and only if there is an index [22Xm ≤ n[122X such that [22Xf_i = g_i[122X for all [22Xi >
  m[122X,  and [22Xtilde{f_m} < tilde{g_m}[122X, where [22Xtilde{c}[122X denotes the integer value in
  [22X{  0,  1,  ...,  p-1  }[122X  that  is  mapped  to  [22Xc ∈ GF(p)[122X under the canonical
  epimorphism that maps the integers onto [22XGF(p)[122X.[133X
  
  [33X[0;0Y[22XΦ_{n,p}[122X  is [13Xprimitive[113X over [22XGF(p)[122X (see [2XIsPrimitivePolynomial[102X ([14X66.4-12[114X)). That
  is,  [22XΦ_{n,p}[122X  is  irreducible,  monic,  and  is  the minimal polynomial of a
  primitive root of [22XGF(p^n)[122X.[133X
  
  [33X[0;0YFor   all   divisors   [22Xd[122X   of   [22Xn[122X   the   compatibility  condition  [22XΦ_{d,p}(
  x^{frac{p^n-1}{p^m-1}}   )  ≡  0  mod  {Φ_{n,p}(x)}[122X  holds.  (That  is,  the
  appropriate  power  of  a zero of [22XΦ_{n,p}[122X is a zero of the Conway polynomial
  [22XΦ_{d,p}[122X.)[133X
  
  [33X[0;0YWith respect to the ordering defined above, [22XΦ_{n,p}[122X shall be minimal.[133X
  
  [33X[0;0YThe  computation of Conway polynomials can be time consuming. Therefore, [5XGAP[105X
  comes  with  a list of precomputed polynomials. If a requested polynomial is
  not  stored  then  [5XGAP[105X  prints  a  warning  and  computes it by checking all
  polynomials  in the order defined above for the defining conditions. If [22Xn[122X is
  not a prime this is probably a very long computation. (Some previously known
  polynomials  with  prime  [22Xn[122X  are  not stored in [5XGAP[105X because they are quickly
  recomputed.)  Use  the function [2XIsCheapConwayPolynomial[102X ([14X59.5-2[114X) to check in
  advance if [2XConwayPolynomial[102X will give a result after a short time.[133X
  
  [33X[0;0YNote  that  primitivity  of  a  polynomial  can  only  be checked if [5XGAP[105X can
  factorize  [22Xp^n-1[122X. A sufficiently new version of the [5XFactInt[105X package contains
  many   precomputed  factors  of  such  numbers  from  various  factorization
  projects.[133X
  
  [33X[0;0YSee [Lüb03] for further information on known Conway polynomials.[133X
  
  [33X[0;0YAn  interactive  overview of the Conway polynomials known to [5XGAP[105X is provided
  by  the  function  [10XBrowseConwayPolynomials[110X  from the [5XGAP[105X package [5XBrowse[105X, see
  [2XBrowseGapData[102X ([14XBrowse: BrowseGapData[114X).[133X
  
  [33X[0;0YIf [3Xpol[103X is a result returned by [2XConwayPolynomial[102X the command [10XPrint( InfoText(
  [3Xpol[103X[10X ) );[110X will print some info on the origin of that particular polynomial.[133X
  
  [33X[0;0YFor  some  purposes it may be enough to have any primitive polynomial for an
  extension   of   a   finite   field   instead   of  the  Conway  polynomial,
  see [2XRandomPrimitivePolynomial[102X ([14X59.5-3[114X) below.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XConwayPolynomial( 2, 5 );  ConwayPolynomial( 3, 7 );[127X[104X
    [4X[28Xx_1^5+x_1^2+Z(2)^0[128X[104X
    [4X[28Xx_1^7-x_1^2+Z(3)^0[128X[104X
  [4X[32X[104X
  
  [1X59.5-2 IsCheapConwayPolynomial[101X
  
  [33X[1;0Y[29X[2XIsCheapConwayPolynomial[102X( [3Xp[103X, [3Xn[103X ) [32X function[133X
  
  [33X[0;0YReturns  [9Xtrue[109X  if  [10XConwayPolynomial( [3Xp[103X[10X, [3Xn[103X[10X )[110X will give a result in [13Xreasonable[113X
  time.  This is either the case when this polynomial is pre-computed, or if [3Xn[103X
  is a not too big prime.[133X
  
  [1X59.5-3 RandomPrimitivePolynomial[101X
  
  [33X[1;0Y[29X[2XRandomPrimitivePolynomial[102X( [3XF[103X, [3Xn[103X[, [3Xi[103X] ) [32X function[133X
  
  [33X[0;0YFor  a  finite  field  [3XF[103X  and  a  positive integer [3Xn[103X this function returns a
  primitive  polynomial  of degree [3Xn[103X over [3XF[103X, that is a zero of this polynomial
  has  maximal multiplicative order [22X|[3XF[103X|^n-1[122X. If [3Xi[103X is given then the polynomial
  is  written  in  variable  number [3Xi[103X over [3XF[103X (see [2XIndeterminate[102X ([14X66.1-1[114X)), the
  default for [3Xi[103X is 1.[133X
  
  [33X[0;0YAlternatively,  [3XF[103X  can  be a prime power q, then [3XF[103X = GF(q) is assumed. And [3Xi[103X
  can  be  a  univariate polynomial over [3XF[103X, then the result is a polynomial in
  the same variable.[133X
  
  [33X[0;0YThis  function  can  work for much larger fields than those for which Conway
  polynomials are available, of course [5XGAP[105X must be able to factorize [22X|[3XF[103X|^n-1[122X.[133X
  
  
  [1X59.6 [33X[0;0YPrinting, Viewing and Displaying Finite Field Elements[133X[101X
  
  [1X59.6-1 ViewObj[101X
  
  [33X[1;0Y[29X[2XViewObj[102X( [3Xz[103X ) [32X method[133X
  [33X[1;0Y[29X[2XPrintObj[102X( [3Xz[103X ) [32X method[133X
  [33X[1;0Y[29X[2XDisplay[102X( [3Xz[103X ) [32X method[133X
  
  [33X[0;0YInternal  finite  field  elements  are  viewed,  printed  and displayed (see
  section  [14X6.3[114X for the distinctions between these operations) as powers of the
  primitive  root  (except for the zero element, which is displayed as 0 times
  the primitive root). Thus:[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XZ(2);[127X[104X
    [4X[28XZ(2)^0[128X[104X
    [4X[25Xgap>[125X [27XZ(5)+Z(5);[127X[104X
    [4X[28XZ(5)^2[128X[104X
    [4X[25Xgap>[125X [27XZ(256);[127X[104X
    [4X[28XZ(2^8)[128X[104X
    [4X[25Xgap>[125X [27XZero(Z(125));[127X[104X
    [4X[28X0*Z(5)[128X[104X
  [4X[32X[104X
  
  [33X[0;0YNote  also  that  each  element  is  displayed as an element of the field it
  generates,  and  that  the  size  of  the field is printed as a power of the
  characteristic.[133X
  
  [33X[0;0YElements  of  larger  fields  are printed as [5XGAP[105X expressions which represent
  them as sums of low powers of the primitive root:[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XPrint( Z(3,20)^100, "\n" );[127X[104X
    [4X[28X2*Z(3,20)^2+Z(3,20)^4+Z(3,20)^6+Z(3,20)^7+2*Z(3,20)^9+2*Z(3,20)^10+2*Z\[128X[104X
    [4X[28X(3,20)^12+2*Z(3,20)^15+2*Z(3,20)^17+Z(3,20)^18+Z(3,20)^19[128X[104X
    [4X[25Xgap>[125X [27XPrint( Z(3,20)^((3^20-1)/(3^10-1)), "\n" );[127X[104X
    [4X[28XZ(3,20)^3+2*Z(3,20)^4+2*Z(3,20)^7+Z(3,20)^8+2*Z(3,20)^10+Z(3,20)^11+2*\[128X[104X
    [4X[28XZ(3,20)^12+Z(3,20)^13+Z(3,20)^14+Z(3,20)^15+Z(3,20)^17+Z(3,20)^18+2*Z(\[128X[104X
    [4X[28X3,20)^19[128X[104X
    [4X[25Xgap>[125X [27XZ(3,20)^((3^20-1)/(3^10-1)) = Z(3,10);[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [33X[0;0YNote  from  the  second  example  above,  that these elements are not always
  written over the smallest possible field before being output.[133X
  
  [33X[0;0YThe  [2XViewObj[102X and [2XDisplay[102X methods for these large finite field elements use a
  slightly  more  compact,  but  mathematically equivalent representation. The
  primitive  root  is  represented by [10Xz[110X; its [22Xi[122X-th power by [10Xz[110X[22Xi[122X and [22Xk[122X times this
  power by [22Xk[122X[10Xz[110X[22Xi[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XZ(5,20)^100;[127X[104X
    [4X[28Xz2+z4+4z5+2z6+z8+3z9+4z10+3z12+z13+2z14+4z16+3z17+2z18+2z19[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThis  output  format is always used for [2XDisplay[102X. For [2XViewObj[102X it is used only
  if  its  length  would  not exceed the number of lines specified in the user
  preference  [10XViewLength[110X  (see  [2XSetUserPreference[102X  ([14X3.2-3[114X).  Longer  output is
  replaced by [10X<<an element of GF([3Xp[103X[10X, [3Xd[103X[10X)>>[110X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XZ(2,409)^100000;[127X[104X
    [4X[28X<<an element of GF(2, 409)>>[128X[104X
    [4X[25Xgap>[125X [27XDisplay(Z(2,409)^100000);[127X[104X
    [4X[28Xz2+z3+z4+z5+z6+z7+z8+z10+z11+z13+z17+z19+z20+z29+z32+z34+z35+z37+z40+z\[128X[104X
    [4X[28X45+z46+z48+z50+z52+z54+z55+z58+z59+z60+z66+z67+z68+z70+z74+z79+z80+z81\[128X[104X
    [4X[28X+z82+z83+z86+z91+z93+z94+z95+z96+z98+z99+z100+z101+z102+z104+z106+z109\[128X[104X
    [4X[28X+z110+z112+z114+z115+z118+z119+z123+z126+z127+z135+z138+z140+z142+z143\[128X[104X
    [4X[28X+z146+z147+z154+z159+z161+z162+z168+z170+z171+z173+z174+z181+z182+z183\[128X[104X
    [4X[28X+z186+z188+z189+z192+z193+z194+z195+z196+z199+z202+z204+z205+z207+z208\[128X[104X
    [4X[28X+z209+z211+z212+z213+z214+z215+z216+z218+z219+z220+z222+z223+z229+z232\[128X[104X
    [4X[28X+z235+z236+z237+z238+z240+z243+z244+z248+z250+z251+z256+z258+z262+z263\[128X[104X
    [4X[28X+z268+z270+z271+z272+z274+z276+z282+z286+z288+z289+z294+z295+z299+z300\[128X[104X
    [4X[28X+z301+z302+z303+z304+z305+z306+z307+z308+z309+z310+z312+z314+z315+z316\[128X[104X
    [4X[28X+z320+z321+z322+z324+z325+z326+z327+z330+z332+z335+z337+z338+z341+z344\[128X[104X
    [4X[28X+z348+z350+z352+z353+z356+z357+z358+z360+z362+z364+z366+z368+z372+z373\[128X[104X
    [4X[28X+z374+z375+z378+z379+z380+z381+z383+z384+z386+z387+z390+z395+z401+z402\[128X[104X
    [4X[28X+z406+z408[128X[104X
  [4X[32X[104X
  
  [33X[0;0YFinally note that elements of large prime fields are stored and displayed as
  residue class objects. So[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XZ(65537);[127X[104X
    [4X[28XZmodpZObj( 3, 65537 )[128X[104X
  [4X[32X[104X
  
