Normal Bases over GF(q),

For q a power of a prime p, it is known that if m is a power of p or m itself is a prime different from p having q as one of its primitive roots, then the roots of any irreducible polynomial of degree m and of non-zero trace are linearly independent over GF(q). As a consequence the roots of such an mth degree polynomial form a basis of GF(qm) over GF(q). Such a basis is called a normal basis over GF(q) and the polynomial is called normal over GF(q). Normal bases over finite fields have proved very useful for fast arithmetic computations with potential applications to coding theory and to cryptography. In this paper, we prove that for mth degree irreducible polynomials the above two conditions are indeed necessary and sufficient conditions for the equivalence between the properties of having a non-zero trace and being normal over GF(q).