// %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
//      The Division algorithm and Euclidean algorithm
//         Polynomial ring in one variable
// %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
// Here are two choices for a field, and the polynomial ring.
// <x> tells magma to use x as the variable.
F := Rationals();
F := GF(2);
P<x> := PolynomialRing(F);


//  Just to  have shorthand:
LT := LeadingTerm;
//  An important piece of magma language is !, the coercion operator.
//  The following means the 0 element of P.
P!0

MyDivAlg := function(f,h)
   q := P!0;
   r := f;
   while r ne 0 and Degree(r) ge Degree(h) do
     t := LT(r) div LT(h);
     q := q + t;
     r := r-t*h;
   end while;
return q,r ;
end function;


MyEuclAlg := function(f,h)
  r := f;
  s := h; 
  while s ne P!0 do
    _,rem := MyDivAlg(r,s);
// The _ is used to throw away the first returned value.
    r := s;
    s := rem;
  end while;
return r;
end function;