Normal bases and irreducible polynomials

Abstract

A normal basis of Fqn over Fq is a basis of the form {α,αq,…,αqn−1}. An irreducible polynomial in Fq[x] is called an N-polynomial if its roots are linearly independent over Fq. Let p be the characteristic of Fq. Pelis et al. showed that every monic irreducible polynomial with degree n and nonzero trace is an N-polynomial provided that n is either a power of p or a prime different from p and q is a primitive root modulo n. Chang et al. proved that the converse is also true. By comparing the number of N-polynomials with that of irreducible polynomials with nonzero traces, we present an alternative treatment to this problem and show that all the results mentioned above can be easily deduced from our main theorem.