SSL ffe.gi Sprache: unbekannt

#############################################################################
##
## This file is part of GAP, a system for computational discrete algebra.
## This file's authors include Werner Nickel, Martin Schönert.
##
## Copyright of GAP belongs to its developers, whose names are too numerous
## to list here. Please refer to the COPYRIGHT file for details.
##
## SPDX-License-Identifier: GPL-2.0-or-later
##
## This file contains methods for `FFE's.
## Note that we must distinguish finite fields and fields that consist of
## `FFE's.
## (The image of the natural embedding of the field `GF(<q>)' into a field
## of rational functions is of course a finite field but its elements are
## not `FFE's since this would be a property given by their family.)
##
## Special methods for (elements of) general finite fields can be found in
## the file `fieldfin.gi'.
##
## The implementation of elements of rings `Integers mod <n>' can be found
## in the file `zmodnz.gi'.
##

#############################################################################
##
#M \+( <ffe>, <rat> )
#M \+( <rat>, <ffe> )
#M \*( <ffe>, <rat> )
#M \*( <rat>, <ffe> )
##
## The arithmetic operations with one operand a FFE <ffe> and the other
## a rational <rat> are defined as follows.
## Let `<one> = One( <ffe> )', and let <num> and <den> denote the numerator
## and denominator of <rat>.
## Let `<new> = (<num>\*<one>) / (<den>\*<one>)'.
## (Note that the multiplication of FFEs with positive integers is defined
## as abbreviated addition.)
## Then we have `<ffe> + <rat> = <rat> + <ffe> = <ffe> + <new>',
## and `<ffe> \* <rat> = <rat> \* <ffe> = <ffe> \* <new>'.
## As usual, difference and quotient are defined as sum and product,
## with the second argument replaced by its additive and multiplicative
## inverse, respectively.
##
## (It would be possible to install these methods in the kernel tables,
## where the case of arithmetic operations with one operand an internally
## represented FFE and the other a rational *integer* is handled.
## But the case of noninteger rationals does probably not occur particularly
## often.)
##
InstallMethod( \+,
 "for a FFE and a rational",
 [ IsFFE, IsRat ],
 function( ffe, rat )
 rat:= (rat mod Characteristic(ffe))*One(ffe);
 return ffe + rat;
 end );

InstallMethod( \+,
 "for a rational and a FFE",
 [ IsRat, IsFFE ],
 function( rat, ffe )
 rat:= (rat mod Characteristic(ffe))*One(ffe);
 return rat + ffe;
 end );

InstallMethod( \*,
 "for a FFE and a rational",
 [ IsFFE, IsRat ],
 function( ffe, rat )
 if IsInt( rat ) then
 # Avoid the recursion trap.
 TryNextMethod();
 fi;
 # Replace the rational by an equivalent integer.
 rat:= rat mod Characteristic(ffe);
 return ffe * rat;
 end );

InstallMethod( \*,
 "for a rational and a FFE",
 [ IsRat, IsFFE ],
 function( rat, ffe )
 if IsInt( rat ) then
 # Avoid the recursion trap.
 TryNextMethod();
 fi;
 # Replace the rational by an equivalent integer.
 rat:= rat mod Characteristic(ffe);
 return rat * ffe;
 end );

#############################################################################
##
#M DegreeFFE( <vector> )
##
InstallOtherMethod( DegreeFFE,
 "for a row vector of FFEs",
 [ IsRowVector and IsFFECollection ],
 function( list )
 local deg, i;

 #
 # Those length zero vectors for which this makes sense have
 # representation-specific methods
 #
 if Length(list) = 0 then
 TryNextMethod();
 fi;
 deg:= DegreeFFE( list[1] );
 for i in [ 2 .. Length( list ) ] do
 deg:= LcmInt( deg, DegreeFFE( list[i] ) );
 od;
 return deg;
 end );
#T list -> Lcm( List( list, DegreeFFE ) ) );
#T to be provided by the kernel!

#############################################################################
##
#M DegreeFFE( <matrix> )
##
InstallOtherMethod( DegreeFFE,
 "for a matrix of FFEs",
 [ IsMatrix and IsFFECollColl ],
 function( mat )
 local deg, i;
 deg:= DegreeFFE( mat[1] );
 for i in [ 2 .. Length( mat ) ] do
 deg:= LcmInt( deg, DegreeFFE( mat[i] ) );
 od;
 return deg;
 end );

#############################################################################
##
#M LogFFE( <n>, <r> ) . . . . . . . . . . . . for two FFE in a prime field
##
InstallMethod( LogFFE,
 "for two FFEs (in a prime field)",
 IsIdenticalObj,
 [ IsFFE, IsFFE ],
 function( n, r )
 if DegreeFFE( n ) = 1 and DegreeFFE( r ) = 1 then
 return LogMod( Int( n ), Int( r ), Characteristic( n ) );
 else
 TryNextMethod();
 fi;
end );

#############################################################################
##
#M IntVecFFE( <vector> )
##
InstallMethod( IntVecFFE,
 "for a row vector of FFEs",
 [ IsRowVector and IsFFECollection ],
 v -> List( v, IntFFE ) );

#############################################################################
##
#F FFEFamily( )
##
InstallGlobalFunction( FFEFamily, function( p )
 local F;

 if MAXSIZE_GF_INTERNAL < p then

 # large characteristic
 F:= GET_FROM_SORTED_CACHE( FAMS_FFE_LARGE, p, function()

 F:= NewFamily( "FFEFamily", IsFFE,
 CanEasilySortElements,
 CanEasilySortElements );
 SetCharacteristic( F, p );

 # Store the type for the representation of prime field elements
 # via residues.
 F!.typeOfZmodnZObj:= NewType( F, IsZmodpZObjLarge
 and IsModulusRep and IsZDFRE);

 SetOne( F, ZmodnZObj( F, 1 ) );
 SetZero( F, ZmodnZObj( F, 0 ) );

 # The whole family is a unique factorisation domain.
 SetFilterObj( F, IsUFDFamily );

 return F;

 end );

 else

 # small characteristic
 # (The list `TYPE_FFE' is used to store the types.)
 F:= FamilyType( TYPE_FFE( p ) );
 if not HasOne( F ) then

 # This family has not been accessed by `FFEFamily' before.
 SetOne( F, One( Z(p) ) );
 SetZero( F, Zero( Z(p) ) );

 fi;

 fi;

 MakeWriteOnceAtomic(F); # needed for HPC-GAP, does nothing in plain GAP
 return F;
end );

#############################################################################
##
#M Zero( <ffe-family> )
##
InstallOtherMethod( Zero,
 "for a family of FFEs",
 [ IsFFEFamily ],
 function( fam )
 local char;
 char:= Characteristic( fam );
 if char <= MAXSIZE_GF_INTERNAL then
 return Zero( Z( char ) );
 else
 TryNextMethod();
 fi;
 end );

#############################################################################
##
#M One( <ffe-family> )
##
InstallOtherMethod( One,
 "for a family of FFEs",
 [ IsFFEFamily ],
 function( fam )
 local char;
 char:= Characteristic( fam );
 if char <= MAXSIZE_GF_INTERNAL then
 return One( Z( char ) );
 else
 TryNextMethod();
 fi;
end );

#############################################################################
##
#F LargeGaloisField( ^<n> )
#F LargeGaloisField( , <n> )
##
#T other construction possibilities?
##

InstallMethod( LargeGaloisField,
 [IsPosInt],
 function(q)
 local p,d;
 p := SmallestRootInt(q);
 d := LogInt(q,p);
 Assert(1, q = p^d);
 Assert(1, IsPrimeInt(p));
 return LargeGaloisField(p,d);
end);

InstallMethod( LargeGaloisField,
 [IsPosInt, IsPosInt],
 function(p,d)
 if not IsPrimeInt(p) then
 Error("LargeGalosField: Characteristic must be prime");
 fi;
 if d = 1 then
 return ZmodpZNC( p );
 else
 TryNextMethod();
 fi;
end );

#############################################################################
##
#F GaloisField( ^<d> ) . . . . . . . . . . create a finite field object
#F GF( ^<d> )
#F GaloisField( , <d> )
#F GF( , <d> )
#F GaloisField( <subfield>, <d> )
#F GF( <subfield>, <d> )
#F GaloisField( , <pol> )
#F GF( , <pol> )
#F GaloisField( <subfield>, <pol> )
#F GF( <subfield>, <pol> )
##

# in Finite field calculations we often ask again and again for the same GF.
# Therefore cache the last entry.
GFCACHE:=[0,0];
MakeThreadLocal("GFCACHE");

InstallGlobalFunction( GaloisField, function ( arg )
 local F, # the field, result
 p, # characteristic
 d, # degree over the prime field
 d1, # degree of subfield over prime field
 q, # size of field to be constructed
 subfield, # left acting domain of the field under construction
 new,
 B; # basis of the extension

 # if necessary split the arguments
 if Length( arg ) = 1 and IsPosInt( arg[1] ) then

 if arg[1]=GFCACHE[1] then
 return GFCACHE[2];
 fi;

 # `GF( p^d )'
 p := SmallestRootInt( arg[1] );
 d := LogInt( arg[1], p );

 elif Length( arg ) = 2 then

 # `GF( p, d )'
 p := arg[1];
 d := arg[2];

 else
 Error( "usage: GF( <subfield>, <extension> )" );
 fi;

 # if the subfield is given by a prime denoting the prime field
 if IsInt( p ) and IsPrimeInt( p ) then

 subfield:= p;

 # if the degree of the extension is given
 if IsInt( d ) and 0 < d then

 # `GF( p, d )' for prime `p'
 if MAXSIZE_GF_INTERNAL < p^d then
 return LargeGaloisField( p, d );
 fi;

 # if the extension is given by an irreducible polynomial
 # over the prime field
 elif IsRationalFunction( d )
 and IsLaurentPolynomial( d )
 and DegreeFFE( CoefficientsOfLaurentPolynomial( d )[1] ) = 1 then

 # `GF( p, <pol> )' for prime `p'
 return FieldExtension( GaloisField( p, 1 ), d );

 # if the extension is given by coefficients of an irred. polynomial
 # over the prime field
 elif IsHomogeneousList( d ) and DegreeFFE( d ) = 1 then

 # `GF( p, <polcoeffs> )' for prime `p'
 return FieldExtension( GaloisField( p, 1 ),
 UnivariatePolynomial( GaloisField(p,1), d ) );

 # if a basis for the extension is given
 elif IsHomogeneousList( d ) then

#T The construction of a field together with a basis is obsolete.
#T One should construct the basis explicitly.
 # `GF( p, <basisvectors> )' for prime `p'
 F := GaloisField( GaloisField( p, 1 ), Length( d ) );

 # Check that the vectors in `d' really form a basis,
 # and construct the basis.
 B:= Basis( F, d );
 if B = fail then
 Error( "<extension> is not linearly independent" );
 fi;

 # Note that `F' is *not* the field stored in the global list!
 SetBasis( F, B );
 return F;

 fi;

 # if the subfield is given by a finite field
 elif IsField( p ) then

 subfield:= p;
 p:= Characteristic( subfield );

 d1 := DegreeOverPrimeField(subfield);
 # if the degree of the extension is given
 if IsInt( d ) then
 q := p^(d*d1);
 if MAXSIZE_GF_INTERNAL < q then
 if d1 = 1 then
 return LargeGaloisField( p, d );
 else
 return FieldByGenerators(subfield, [Z(p,d*d1)]);
 fi;
 fi;

 d:= d * DegreeOverPrimeField( subfield );

 # if the extension is given by coefficients of an irred. polynomial
#T should be obsolete!
 elif IsHomogeneousList( d )
 and DegreeOverPrimeField( subfield ) mod DegreeFFE( d ) = 0 then

 # `GF( subfield, <polcoeffs> )'
 return FieldExtension( subfield,
 UnivariatePolynomial( subfield, d ) );

 # if the extension is given by an irreducible polynomial
 elif IsRationalFunction( d )
 and IsLaurentPolynomial( d )
 and DegreeOverPrimeField( subfield ) mod
 DegreeFFE( CoefficientsOfLaurentPolynomial( d )[1] ) = 0 then

 # `GF( subfield, <pol> )'
 return FieldExtension( subfield, d );

 # if a basis for the extension is given
#T The construction of a field together with a basis is obsolete.
 elif IsHomogeneousList( d ) then

 # `GF( <subfield>, <basisvectors> )'
 F := GaloisField( subfield, Length( d ) );

 # Check that the vectors in `d' really form a basis,
 # and construct the basis.
 B:= Basis( F, d );
 if B = fail then
 Error( "<extension> is not linearly independent" );
 fi;

 # Note that `F' is *not* the field stored in the global list!
 SetBasis( F, B );
 return F;

 # Otherwise we don't know how to handle the extension.
 else
 Error( "<extension> must be a <deg>, <bas>, or <pol>" );
 fi;

 # Otherwise we don't know how to handle the subfield.
 else
 Error( "<subfield> must be a prime or a finite field" );
 fi;

 # If this place is reached,
 # `p' is the characteristic, `d' is the degree of the extension,
 # and `p^d' is less than or equal to `MAXSIZE_GF_INTERNAL'.

 if IsInt( subfield ) then

 # The standard field is required. Look whether it is already stored.

 new := false;
 F := GET_FROM_SORTED_CACHE(GALOIS_FIELDS, p^d, function()
 new := true;
 # Construct the finite field object.
 if d = 1 then
 F:= FieldOverItselfByGenerators( [ Z(p) ] );
 else
 F:= FieldByGenerators( FieldOverItselfByGenerators( [ Z(p) ] ),
 [ Z(p^d) ] );
 fi;
 return F;
 end );

 if Length(arg)=1 and not new then
 GFCACHE:=[arg[1],F];
 fi;

 else

 # Construct the finite field object.
 F:= FieldByGenerators( subfield, [ Z(p^d) ] );

 fi;

 # Return the finite field.
 return F;
end );

#############################################################################
##
#M FieldExtension( <subfield>, <poly> )
##
InstallOtherMethod( FieldExtension,
 "for a field of FFEs, and a univ. Laurent polynomial",
#T CollPoly
 [ IsField and IsFFECollection, IsLaurentPolynomial ],
 function( F, poly )

 local coeffs, p, d, z, r, one, zero, E;

 coeffs:= CoefficientsOfLaurentPolynomial( poly );
 coeffs:= ShiftedCoeffs( coeffs[1], coeffs[2] );
 p:= Characteristic( F );
 d:= ( Length( coeffs ) - 1 ) * DegreeOverPrimeField( F );

 if MAXSIZE_GF_INTERNAL < p^d then
 TryNextMethod();
 fi;

 # Compute a root of the defining polynomial.
 z := Z( p^d );
 r := z;
 one:= One( r );
 zero:= Zero( r );
 while r <> one and ValuePol( coeffs, r ) <> zero do
 r := r * z;
 od;
 if DegreeFFE( r ) < Length( coeffs ) - 1 then
 Error( "<poly> must be irreducible" );
 fi;

 # We must not call `AsField' here because then the standard `GF(p^d)'
 # would be returned whenever `F' is equal to `GF(p)'.
 E:= FieldByGenerators( F, [ z ] );
 SetDefiningPolynomial( E, poly );
 SetRootOfDefiningPolynomial( E, r );
 if r = z or Order( r ) = Size( E ) - 1 then
 SetPrimitiveRoot( E, r );
 else
 SetPrimitiveRoot( E, z );
 fi;

 return E;
 end );

#############################################################################
##
#M DefiningPolynomial( <F> ) . . . . . . . . . . for standard finite fields
##
InstallMethod( DefiningPolynomial,
 "for a field of FFEs",
 [ IsField and IsFFECollection ],
 function( F )
 local root;

 if HasRootOfDefiningPolynomial( F ) then
 # We must choose a compatible polynomial.
 return MinimalPolynomial( LeftActingDomain( F ),
 RootOfDefiningPolynomial( F ) );
 fi;

 # Choose a primitive polynomial, and store a root.
 root:= Z( Size( F ) );
 SetRootOfDefiningPolynomial( F, root );
 if IsPrimeField( LeftActingDomain( F ) ) then
 return ConwayPolynomial( Characteristic( F ),
 DegreeOverPrimeField( F ) );
 else
 return MinimalPolynomial( LeftActingDomain( F ), root );
 fi;
 end );

#############################################################################
##
#M RootOfDefiningPolynomial( <F> ) . . . . . . . for standard finite fields
##
InstallMethod( RootOfDefiningPolynomial,
 "for a small field of FFEs",
 [ IsField and IsFFECollection ],
 function( F )
 local coeffs, p, d, z, r, one, zero;

 coeffs:= CoefficientsOfLaurentPolynomial( DefiningPolynomial( F ) );

 # Maybe the call to `DefiningPolynomial' has caused that a root is bound.
 if HasRootOfDefiningPolynomial( F ) then
 return RootOfDefiningPolynomial( F );
 fi;

 coeffs:= ShiftedCoeffs( coeffs[1], coeffs[2] );
 p:= Characteristic( F );
 d:= DegreeOverPrimeField( F );

 if Length( coeffs ) = 2 then
 return - coeffs[1] / coeffs[2];
 elif MAXSIZE_GF_INTERNAL < p^d then
 TryNextMethod();
 fi;

 # Compute a root of the defining polynomial.
 z := Z( p^d );
 r := z;
 one:= One( r );
 zero:= Zero( r );
 while r <> one and ValuePol( coeffs, r ) <> zero do
 r := r * z;
 od;
 if DegreeFFE( r ) < Length( coeffs ) - 1 then
 Error( "<poly> must be irreducible" );
 fi;

 # Return the root.
 return r;
 end );

#############################################################################
##
#M ViewObj( <F> ) . . . . . . . . . . . . . . . . . . view a field of `FFE's
#M PrintObj( <F> ) . . . . . . . . . . . . . . . . . print a field of `FFE's
#M String( <F> ) . . . . . . . . . . a string representing a field of `FFE's
#M ViewString( <F> ) . . . . . a short string representing a field of `FFE's
##

GAPInfo.tmpGFstring := function( F )
 if IsPrimeField( F ) then
 return Concatenation( "GF(", String(Characteristic( F )), ")" );
 elif IsPrimeField( LeftActingDomain( F ) ) then
 return Concatenation( "GF(", String(Characteristic( F )),
 "^", String(DegreeOverPrimeField( F )), ")" );
 elif F = LeftActingDomain( F ) then
 return Concatenation( "FieldOverItselfByGenerators( ",
 String(GeneratorsOfField( F )), " )" );
 else
 return Concatenation( "AsField( ", String(LeftActingDomain( F )),
 ", GF(", String(Characteristic( F )),
 "^", String(DegreeOverPrimeField( F )), ") )" );
 fi;
end;
InstallMethod( String, "for a field of FFEs",
 [ IsField and IsFFECollection ], 10, GAPInfo.tmpGFstring );

InstallMethod( ViewString, "for a field of FFEs",
 [ IsField and IsFFECollection ], 10, GAPInfo.tmpGFstring );
Unbind(GAPInfo.tmpGFstring);
InstallMethod( ViewObj, "for a field of FFEs",
 [ IsField and IsFFECollection ], 10, function( F )
 Print( ViewString(F) );
end );

InstallMethod( PrintObj, "for a field of FFEs",
 [ IsField and IsFFECollection ], 10, function( F )
 Print( ViewString(F) );
end );

#############################################################################
##
#M \in( <z> ,<F> ) . . . . . . . . test if an object lies in a finite field
##
InstallMethod( \in,
 "for a FFE, and a field of FFEs",
 IsElmsColls,
 [ IsFFE, IsField and IsFFECollection ],
 function ( z, F )
 return DegreeOverPrimeField( F ) mod DegreeFFE( z ) = 0;
 end );

#############################################################################
##
#M Intersection( <F>, <G> ) . . . . . . . intersection of two finite fields
##
InstallMethod( Intersection2,
 "for two fields of FFEs",
 IsIdenticalObj,
 [ IsField and IsFFECollection, IsField and IsFFECollection ],
 function ( F, G )
 return GF( Characteristic( F ), GcdInt( DegreeOverPrimeField( F ),
 DegreeOverPrimeField( G ) ) );
 end );

#############################################################################
##
#M Conjugates( <L>, <K>, <z> ) . . . . conjugates of a finite field element
##
InstallMethod( Conjugates,
 "for two fields of FFEs, and a FFE",
 IsCollsXElms,
 [ IsField and IsFinite and IsFFECollection,
 IsField and IsFinite and IsFFECollection, IsFFE ],
 function( L, K, z )
 local cnjs, # conjugates of <z> in <L>/<K>, result
 ord, # order of the subfield <K>
 deg, # degree of <L> over <K>
 i; # loop variable

 if DegreeOverPrimeField( L ) mod DegreeFFE(z) <> 0 then
 Error( "<z> must lie in <L>" );
 fi;

 # Get the order of `K' and the dimension of `L' as a `K'-vector space.
 ord := Size( K );
 deg := DegreeOverPrimeField( L ) / DegreeOverPrimeField( K );

 # compute the conjugates $\set_{i=0}^{d-1}{z^(q^i)}$
 cnjs := [];
 for i in [0..deg-1] do
 Add( cnjs, z );
 z := z^ord;
 od;

 # return the conjugates
 return cnjs;
 end );

#############################################################################
##
#F Norm( <L>, <K>, <z> ) . . . . . . . . . norm of a finite field element
##
InstallMethod( Norm,
 "for two fields of FFEs, and a FFE",
 IsCollsXElms,
 [ IsField and IsFinite and IsFFECollection,
 IsField and IsFinite and IsFFECollection, IsFFE ],
 function( L, K, z )

 if DegreeOverPrimeField( L ) mod DegreeFFE(z) <> 0 then
 Error( "<z> must lie in <L>" );
 fi;

 # Let $|K| = q$, $|L| = q^d$.
 # The norm of $z$ is
 # $\prod_{i=0}^{d-1} (z^{q^i}) = z^{\sum_{i=0}^{d-1} q^i}
 # = z^{\frac{q^d-1}{q-1}$.
 return z ^ ( ( Size(L) - 1 ) / ( Size(K) - 1 ) );
 end );

#############################################################################
##
#M Trace( <L>, <K>, <z> ) . . . . . . . . . trace of a finite field element
##
InstallMethod( Trace,
 "for two fields of FFEs, and a FFE",
 IsCollsXElms,
 [ IsField and IsFinite and IsFFECollection,
 IsField and IsFinite and IsFFECollection, IsFFE ],
 function( L, K, z )
 local trc, # trace of <z> in <L>/<K>, result
 ord, # order of the subfield <K>
 deg, # degree of <L> over <K>
 i; # loop variable

 if DegreeOverPrimeField( L ) mod DegreeFFE(z) <> 0 then
 Error( "<z> must lie in <L>" );
 fi;

 # Get the order of `K' and the dimension of `L' as a `K'-vector space.
 ord := Size( K );
 deg := DegreeOverPrimeField( L ) / DegreeOverPrimeField( K );

 # $trc = \sum_{i=0}^{deg-1}{ z^(ord^i) }$
 trc := 0;
 for i in [0..deg-1] do
 trc := trc + z;
 z := z^ord;
 od;

 # return the trace
 return trc;
 end );

#############################################################################
##
#M Order( <z> ) . . . . . . . . . . . . . . order of a finite field element
##
InstallMethod( Order,
 "for an internal FFE",
 [ IsFFE and IsInternalRep ],
 function ( z )
 local ord, # order of <z>, result
 chr, # characteristic of <F> (and <z>)
 deg; # degree of <z> over the prime field

 # compute the order
 if IsZero( z ) then
 ord := 0;
 else
 chr := Characteristic( z );
 deg := DegreeFFE( z );
 ord := (chr^deg-1) / GcdInt( chr^deg-1, LogFFE( z, Z(chr^deg) ) );
 fi;

 # return the order
 return ord;
end );

InstallMethod( Order,
 "for a general FFE",
 [IsFFE],
 function(z)
 local p, d, ord, facs, f, i, o;
 p := Characteristic(z);
 d := DegreeFFE(z);
 ord := p^d-1;
 facs := Collected(Factors(Integers,ord));
 for f in facs do
 for i in [1..f[2]] do
 o := ord/f[1];
 if not IsOne(z^o) then
 break;
 fi;
 ord := o;
 od;
 od;
 return ord;
end);

#############################################################################
##
#M SquareRoots( <F>, <z> )
##
InstallMethod( SquareRoots,
 "for a field of FFEs, and a FFE",
 IsCollsElms,
 [ IsField, IsFFE ],
 function( F, z )
 local r;
 if IsZero( z ) then
 return [ z ];
 elif Characteristic( z ) = 2 then

 # unique square root for each element
 r:= PrimitiveRoot( F );
 return [ r ^ ( LogFFE( z, r ) / 2 mod ( Size( F )-1 ) ) ];

 else

 # either two solutions in `F' or no solution
 r:= PrimitiveRoot( F );
 z:= LogFFE( z, r ) / 2;
 if IsInt( z ) then
 z:= r ^ z;
 return Set( [ z, -z ] );
 else
 return [];
 fi;

 fi;
 end );

#############################################################################
##
#M NthRoot( <F>, <z>, <n> )
##
InstallMethod( NthRoot, "for a field of FFEs, and a FFE", IsCollsElmsX,
 [ IsField, IsFFE,IsPosInt ],
function( F, a,n )
local z,qm;
 if IsOne(a) or IsZero(a) or n=1 then
 return a;
 fi;
 z:=PrimitiveRoot(F);
 qm:=Size(F)-1;
 a:=LogFFE(a,z)/n;
 if 1<GcdInt(DenominatorRat(a),qm) then
 return fail;
 fi;
 return z^(a mod qm);
end);

#############################################################################
##
#M Int( <z> ) . . . . . . . . . convert a finite field element to an integer
##
InstallMethod( Int,
 "for an FFE",
 [ IsFFE ],
 IntFFE );

#############################################################################
##
#M IntFFESymm( <z> )
##
InstallMethod(IntFFESymm,"FFE",true,[ IsFFE ],0,
function(z)
local i,p;
 p:=Characteristic(z);
 i:=IntFFE(z);
 if 2*i>p then
 return i-p;
 else
 return i;
 fi;
end);

#############################################################################
##
#M IntFFESymm( <vector> )
##
InstallOtherMethod(IntFFESymm,"vector",true,
 [IsRowVector and IsFFECollection ],0,
 v -> List( v, IntFFESymm ) );

#############################################################################
##
#M String( <ffe> ) . . . . . . convert a finite field element into a string
##
InstallMethod(String,"for an internal FFE",true,[IsFFE and IsInternalRep],0,
function ( ffe )
local str, log,deg,char;
 char:=Characteristic(ffe);
 if IsZero( ffe ) then
 str := Concatenation("0*Z(",String(char),")");
 else
 str := Concatenation("Z(",String(char));
 deg:=DegreeFFE(ffe);
 if deg <> 1 then
 str := Concatenation(str,"^",String(deg));
 fi;
 str := Concatenation(str,")");
 log:= LogFFE(ffe,Z( char ^ deg ));
 if log <> 1 then
 str := Concatenation(str,"^",String(log));
 fi;
 fi;
 ConvertToStringRep( str );
 return str;
end );

InstallMethod(ViewString, "for an internal FFE delegating to String",
 [IsFFE and IsInternalRep], String );

InstallMethod(DisplayString, "for an internal FFE via String",
 [IsFFE and IsInternalRep], ffe -> Concatenation( String(ffe), "\n") );

#############################################################################
##
#M FieldOverItselfByGenerators( <elms> )
##
InstallMethod( FieldOverItselfByGenerators,
 "for a collection of FFEs",
 [ IsFFECollection ],
 function( elms )

 local F, d, q;

 F:= Objectify( NewType( FamilyObj( elms ),
 IsField and IsAttributeStoringRep ),
 rec() );
 d:= DegreeFFE( elms );
 q:= Characteristic( F )^d;

 SetLeftActingDomain( F, F );
 SetIsPrimeField( F, d = 1 );
 SetIsFinite( F, true );
 SetSize( F, q );
 SetGeneratorsOfDivisionRing( F, elms );
 SetGeneratorsOfRing( F, elms );
 SetDegreeOverPrimeField( F, d );
 SetDimension( F, 1 );

 if q <= MAXSIZE_GF_INTERNAL then
 SetPrimitiveRoot( F, Z(q) );
 fi;

 return F;
 end );

#############################################################################
##
#M FieldByGenerators( <F>, <elms> ) . . . . . . . . . . field by generators
##
InstallMethod( FieldByGenerators,
 "for two coll. of FFEs, the first a field",
 IsIdenticalObj,
 [ IsFFECollection and IsField, IsFFECollection ],
 function( subfield, gens )

 local F, d, subd, q, z;

 F := Objectify( NewType( FamilyObj( gens ),
 IsField and IsAttributeStoringRep ),
 rec() );

 d:= DegreeFFE( gens );
 subd:= DegreeOverPrimeField( subfield );
 if d mod subd <> 0 then
 d:= LcmInt( d, subd );
 gens:= Concatenation( gens, GeneratorsOfDivisionRing( subfield ) );
 fi;

 q:= Characteristic( subfield )^d;

 SetLeftActingDomain( F, subfield );
 SetIsPrimeField( F, d = 1 );
 SetIsFinite( F, true );
 SetSize( F, q );
 SetDegreeOverPrimeField( F, d );
 SetDimension( F, d / DegreeOverPrimeField( subfield ) );

 if q <= MAXSIZE_GF_INTERNAL then
 z:= Z(q);
 SetPrimitiveRoot( F, z );
 gens:= [ z ];
# elif d <> 1 then
# Error( "sorry, large non-prime fields are not yet implemented" );
 fi;

 SetGeneratorsOfDivisionRing( F, gens );
 SetGeneratorsOfRing( F, gens );

 return F;
 end );

#############################################################################
##
#M DefaultFieldByGenerators( <z> ) . . . . . . default field containing ffes
#M DefaultFieldByGenerators( <F>, <elms> ) . . default field containing ffes
##
InstallMethod( DefaultFieldByGenerators,
 "for a collection of FFEs that is a list",
 [ IsFFECollection and IsList ],
 gens -> GF( Characteristic( gens ), DegreeFFE( gens ) ) );

InstallOtherMethod( DefaultFieldByGenerators,
 "for a finite field, and a collection of FFEs that is a list",
 IsIdenticalObj,
 [ IsField and IsFinite, IsFFECollection and IsList ],
 function( F, gens )
 return GF( F, DegreeFFE( gens ) );
 end );

#############################################################################
##
#M RingByGenerators( <elms> ) . . . . . . . . . . . . . for FFE collection
#M RingWithOneByGenerators( <elms> ) . . . . . . . . . . for FFE collection
#M DefaultRingByGenerators( <z> ) . . . . . . default ring containing FFEs
#M FLMLORByGenerators( <F>, <elms> ) . . . . . . . . . . for FFE collection
#M FLMLORWithOneByGenerators( <F>, <elms> ) . . . . . . for FFE collection
##
## In all these cases, the result is either zero or in fact a field,
## so we may delegate to `GF'.
##
BindGlobal( "RingFromFFE", function( gens )
 local F;

 F:= GF( Characteristic( gens ), DegreeFFE( gens ) );
 if ForAll( gens, IsZero ) then
 F:= TrivialSubalgebra( F );
 fi;
 return F;
end );

InstallMethod( RingByGenerators,
 "for a collection of FFE",
 [ IsFFECollection ],
 RingFromFFE );

InstallMethod( RingWithOneByGenerators,
 "for a collection of FFE",
 [ IsFFECollection ],
 RingFromFFE );

InstallMethod( DefaultRingByGenerators,
 "for a collection of FFE",
 [ IsFFECollection and IsList ],
 RingFromFFE );

BindGlobal( "FLMLORFromFFE", function( F, elms )
 if ForAll( elms, IsZero ) then
 return TrivialSubalgebra( F );
 else
 return GF( Characteristic( F ),
 Lcm( DegreeFFE( elms ), DegreeOverPrimeField( F ) ) );
 fi;
end );

InstallMethod( FLMLORByGenerators,
 "for a field, and a collection of FFE",
 IsIdenticalObj,
 [ IsField and IsFFECollection, IsFFECollection ], 0,
 FLMLORFromFFE );

InstallMethod( FLMLORWithOneByGenerators,
 "for a field, and a collection of FFE",
 IsIdenticalObj,
 [ IsField and IsFFECollection, IsFFECollection ], 0,
 FLMLORFromFFE );

#############################################################################
##
#M IsGeneratorsOfMagmaWithInverses( <ffelist> )
##
InstallMethod( IsGeneratorsOfMagmaWithInverses,
 "for a collection of FFEs",
 [ IsFFECollection ],
 ffelist -> ForAll( ffelist, x -> not IsZero( x ) ) );

#############################################################################
##
#M AsInternalFFE( <internal ffe> )
##
InstallMethod( AsInternalFFE, [IsFFE and IsInternalRep],
 x->x);

#############################################################################
##
#M AsInternalFFE( <non-ffe> )
##
InstallOtherMethod( AsInternalFFE, [IsObject],
 function(x)
 if not IsFFE(x) then
 return fail;
 else
 TryNextMethod();
 fi;
end);

#############################################################################
##
#M RootFFE( <z>, <k> )
##
InstallMethod(RootFFE,"use field order",IsCollsElmsX,
 [IsField,IsFFE,IsPosInt],0,
function(F,z,k)
 return RootFFE(Size(F),z,k);
end);

InstallOtherMethod(RootFFE,"use LogFFE",true,[IsPosInt,IsFFE,IsPosInt],
function(q,z,k)
local e,m,p,a;
 if IsZero(z) or IsOne(z) then return z;fi;
 m:=q-1;
 e:=LogFFE(z,Z(q));
 p:=GcdInt(m,e); #power for subgroup it is in
 k:=k mod m;
 a:=GcdInt(m,k); # power for subgroup we want
 if p mod a<>0 then
 return fail; # element does not lie in subgroup of a-th powers
 fi;
 # k/a is coprime to order of elts in subgroup and elt is an a-th power
 a:=(e*(a/k mod (m/p))/a mod m);
 return Z(q)^a;
end);

InstallOtherMethod(RootFFE,"without field",true,
 [IsFFE,IsPosInt],0,
function(z,k)
 return RootFFE(Characteristic(z)^DegreeFFE(z),z,k);
end);

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