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#############################################################################
##
#A StandardFF.gd Frank Lübeck
##
## The files StandardFF.g{i,d} contain code to compute standard models of
## finite fields. Fields are returned as field towers with prime degree
## indices, and as single extensions over the prime field.
##
# standard finite fields as towers of prime degree extensions
DeclareGlobalFunction("SteinitzNumberForPrimeDegree");
DeclareGlobalFunction("SteinitzNumberForPrimeDegreeNTL");
DeclareGlobalFunction("StandardPrimeDegreePolynomial");
DeclareGlobalFunction("_ExtensionWithTowerBasis");
# standard finite fields as simple extensions over prime field GF(p)
DeclareGlobalFunction("StandardFiniteField");
DeclareSynonym("FF", StandardFiniteField);
DeclareProperty("IsStandardFiniteField",
IsField and IsFinite and IsAlgebraicExtension);
DeclareCategory("IsStandardFiniteFieldElement",
IsAlgebraicElement and IsKroneckerConstRep and
IsNoImmediateMethodsObject);
DeclareAttribute("PrimitivePowersInTowerBasis", IsStandardFiniteField);
DeclareAttribute("TowerBasis", IsStandardFiniteField);
DeclareOperation("SteinitzNumber", [IsStandardFiniteField, IsRingElement]);
DeclareOperation("ElementSteinitzNumber", [IsStandardFiniteField, IsInt]);
DeclareAttribute("AsVector", IsStandardFiniteFieldElement);
DeclareAttribute("GeneratorMonomials", IsStandardFiniteField);
DeclareAttribute("TowerBasisMonomials", IsStandardFiniteField);
DeclareAttribute("AsPolynomial", IsStandardFiniteFieldElement);
DeclareOperation("ElementVector", [IsStandardFiniteField, IsRowVector]);
DeclareOperation("ElementPolynomial", [IsStandardFiniteField, IsPolynomial]);
# embeddings
DeclareGlobalFunction("StdMon");
DeclareGlobalFunction("StdMonDegs");
DeclareGlobalFunction("StdMonMap");
DeclareGlobalFunction("EmbedSteinitz");
DeclareCategory("IsStandardFiniteFieldEmbedding", IsFieldHomomorphism);
DeclareOperation("SteinitzPair", [IsStandardFiniteFieldElement]);
DeclareAttribute("SteinitzPairConwayGenerator", IsStandardFiniteField);
DeclareGlobalFunction("StandardIsomorphismGF");
# helper to identify GF(p)'s in method selection
DeclareProperty("IsStandardPrimeField", IsField and IsFinite);
# creating elements
DeclareOperation("ZZ", [IsInt, IsInt, IsList]);
# arithmetic in algebraic closure
DeclareGlobalFunction("MoveToCommonStandardField");
DeclareGlobalFunction("MoveToSmallestStandardField");
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