Existence and Cardinality of -Normal Elements in Finite Fields
arXiv:2003.09748 · doi:10.1007/978-3-030-68869-1_15
Abstract
Normal bases in finite fields constitute a vast topic of large theoretical and practical interest. Recently, -normal elements were introduced as a natural extension of normal elements. The existence and the number of -normal elements in a fixed extension of a finite field are both open problems in full generality, and comprise a promising research avenue. In this paper, we first formulate a general lower bound for the number of -normal elements, assuming that they exist. We further derive a new existence condition for -normal elements using the general factorization of the polynomial into cyclotomic polynomials. Finally, we provide an existence condition for normal elements in $\fqm$ with a non-maximal but high multiplicative order in the group of units of the finite field.