Existence and properties of k-normal elements over finite fields

An element α∈Fqnα∈Fqn is normal   over FqFq if {α,αq,…,αqn−1}{α,αq,…,αqn−1} is a basis for FqnFqn over FqFq. It is well known that α∈Fqnα∈Fqn is normal over FqFq if and only if the polynomials gα(x)=αxn−1+αqxn−2+⋯+αqn−2x+αqn−1gα(x)=αxn−1+αqxn−2+⋯+αqn−2x+αqn−1 and xn−1xn−1 are relatively prime over FqnFqn, that is, the degree of their greatest common divisor in Fqn[x]Fqn[x] is 0. An element α∈Fqnα∈Fqn is k-normal   over FqFq if the greatest common divisor of the polynomials gα(x)gα(x) and xn−1xn−1 in Fqn[x]Fqn[x] has degree k; so an element which is normal in the usual sense is 0-normal. In this paper, we introduce and characterize k-normal elements, establish a formula and numerical bounds for the number of k-normal elements and prove an existence result for primitive 1-normal elements.