Communications Toolbox

Characterization of Polynomials

Given a polynomial over GF(p), the gfprimck function determines whether it is irreducible and/or primitive. By definition, if it is primitive then it is irreducible; however, the reverse is not necessarily true. The gfprimdf and gfprimfd functions return primitive polynomials.

Given an element of GF(p^m), the gfminpol function computes its minimal polynomial over GF(p).

Example

For example, the code below reflects the irreducibility of all minimal polynomials. However, the minimal polynomial of a nonprimitive element is not a primitive polynomial.

• p = 3; m = 4;
  % Use default primitive polynomial here.
  
  prim_poly = gfminpol(1,m,p);
  ckprim = gfprimck(prim_poly,p);
  % ckprim = 1, since prim_poly represents a primitive polynomial.
  
  notprimpoly = gfminpol(5,m,p);
  cknotprim = gfprimck(notprimpoly,p);
  % cknotprim = 0 (irreducible but not primitive)
  % since alpha^5 is not a primitive element when p = 3.
  
  ckreducible = gfprimck([0 1 1],p);
  % ckreducible = -1 since the polynomial is reducible.
  

Polynomial Arithmetic

Roots of Polynomials