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#############################################################################
##
#W upoly.gd GAP Library Alexander Hulpke
##
#H @(#)$Id: upoly.gd,v 4.26 2002/04/15 10:05:25 sal Exp $
##
#Y Copyright (C) 1997, Lehrstuhl D fuer Mathematik, RWTH Aachen, Germany
#Y (C) 1999 School Math and Comp. Sci., University of St. Andrews, Scotland
#Y Copyright (C) 2002 The GAP Group
##
## This file contains attributes, properties and operations for univariate
## polynomials
##
Revision.upoly_gd:=
"@(#)$Id: upoly.gd,v 4.26 2002/04/15 10:05:25 sal Exp $";
#############################################################################
##
#A SplittingField(<f>)
##
## returns the smallest field which contains the coefficients of <f> and
## the roots of <f>.
DeclareAttribute("SplittingField",IsPolynomial);
#############################################################################
##
#A IrrFacsPol( <f> ) . . . lists of irreducible factors of polynomial over
## diverse rings
##
## is used to store irreducible factorizations of the polynomial <f>.
## The values of this attribute are lists of the form
## `[ [ <R>, <factors> ], ... ]' where <factors> is
## a list of the irreducible factors of <f> over the coefficients ring <R>.
##
DeclareAttribute("IrrFacsPol",IsPolynomial,"mutable");
#############################################################################
##
#F StoreFactorsPol( <pring>, <upol>, <factlist> ) . . . . store factors list
##
DeclareGlobalFunction("StoreFactorsPol");
#############################################################################
##
#O FactorsSquarefree( <pring>, <upol>, <opt> )
##
## returns a factorization of the squarefree, monic, univariate polynomial
## <upoly> in the polynomial ring <pring>;
## <opt> must be a (possibly empty) record of options.
## <upol> must not have zero as a root.
## This function is used by the factoring algorithms.
##
DeclareOperation("FactorsSquarefree",[IsPolynomialRing,
IsUnivariatePolynomial, IsRecord ]);
#############################################################################
##
#F RootsOfUPol(<upol>)
#F RootsOfUPol(<field>,<upol>)
#F RootsOfUPol("split",<upol>)
##
## This function returns a list of all roots of the univariate polynomial
## <upol> in its default domain. If <field> is given the roots over <field>
## are taken, if the first parameter is the string `"split"' the field is
## taken to be the splitting field of the polynomial.
##
DeclareGlobalFunction("RootsOfUPol");
#############################################################################
##
#V CYCLOTOMICPOLYNOMIALS . . . . . . . . . . list of cyclotomic polynomials
##
## global list encoding cyclotomic polynomials by their coefficients lists
##
DeclareGlobalVariable( "CYCLOTOMICPOLYNOMIALS",
"list, at position n the coefficient list of the n-th cycl. pol." );
InstallFlushableValue( CYCLOTOMICPOLYNOMIALS, [] );
#############################################################################
##
#F CyclotomicPol( <n> ) . . . coefficients of <n>-th cyclotomic polynomial
##
## is the coefficients list of the <n>-th cyclotomic polynomial over
## the rationals.
##
DeclareGlobalFunction( "CyclotomicPol" );
#############################################################################
##
#F CyclotomicPolynomial( <F>, <n> ) . . . . . . <n>-th cycl. pol. over <F>
##
## is the <n>-th cyclotomic polynomial over the ring <F>.
##
DeclareGlobalFunction( "CyclotomicPolynomial" );
#############################################################################
##
#O IsPrimitivePolynomial( <F>, <pol> )
##
## For a univariate polynomial <pol> of degree $d$ in the indeterminate $X$,
## with coefficients in a finite field <F> with $q$ elements, say,
## `IsPrimitivePolynomial' returns `true' if
## \beginlist%ordered
## \item{1.}
## <pol> divides $X^{q^d-1} - 1$, and
## \item{2.}
## for each prime divisor $p$ of $q^d - 1$, <pol> does not divide
## $X^{(q^d-1)/p} - 1$,
## \endlist
## and `false' otherwise.
##
DeclareOperation( "IsPrimitivePolynomial", [ IsField, IsRationalFunction ] );
#############################################################################
##
#F CompanionMat( <poly> )
##
## computes a companion matrix of the polynomial <poly>. This matrix has
## <poly> as its minimal polynomial.
DeclareGlobalFunction( "CompanionMat" );
#############################################################################
##
#F AllIrreducibleMonicPolynomials( <degree>, <field> )
##
DeclareGlobalFunction( "AllIrreducibleMonicPolynomials" );
#############################################################################
##
#E
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