Normal Elements and Irreducible Polynomials in Finite Fields
Let Fq be the finite field of q elements,and Fqn be its extension of degree n.An element α e Fqn is called a normal element of Fqn/Fqif {α,αq,...,αqn-1 }constitutes a basis of Fqn/Fq.Normal elements over finite fields have proved very useful for fast arithmetic computations with potential applications to coding theory and to cryptography.The minimal polynomial of a normal element is certainly an irreducible polynomial with nonzero trace,while the converse does not hold in general.Using linearized polynomials,we give some necessary and sufficient conditions for this problem,which extend the known results.