##################################################
##
##
##    Assumes f is defined over GF(q)
##    and returns f mod an irred poly 
##    (irreducible in GF(q)[x]), as a element of GF(q^d)
##
##
##
##
##
##################################################

HomToFF:=function(f,irred_poly)
# assumed f is defined over F=GF(q)
# returns f mod irred_poly (irreducible in F[x])
# as a element of F
# below FF=F[x]/(f(x))
local FF,as,a,q,p;
F:=DefaultField(CoefficientsOfUnivariatePolynomial(f));
FF:=GF(F,irred_poly);
as:=RootsOfUPol(FF,irred_poly);
a:=Minimum(as);
return Value(f,a);
end;

#################### Examples #####################


x:=X(GF(9),"x");
f:=x^7+x^3+x+1;
DefaultField(CoefficientsOfUnivariatePolynomial(f));
as:=RootsOfUPol(GF(3^7),f);
a:=Minimum(as);
pol:=x^2+x+1;
Value(pol,a);
HomToFF(pol,f);