Existence and properties of k-normal elements over finite fields

An element α^Fqn is normal over F_q if {α,^αq,.,αqn-1} is a basis for ^Fqn over F_q. It is well known that α^Fqn is normal over F _q if and only if the polynomials ^gα(x)=αxn- ¹+^αqxn-²+â̄+αqn- 2x+αqn-1 and ^xn-1 are relatively prime over ^Fqn, that is, the degree of their greatest common divisor in ^Fqn[x] is 0. An element α^Fqn is k-normal over F_q if the greatest common divisor of the polynomials ^gα(x) and ^xn-1 in ^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.