Existência e propriedades de elementos k-normais em corpos finitos

Abstract

This work consists of a detailed study of the k-normal elements in a given finite field and their properties. In it is presented the definition of k-normal element given initially by the authors S. Huczynska, G. Mullen, D. Panario, D. Thomson in cite{artiprinci}, which generalizes the notion of normal element. It also shows four theorems of great impact given in \cite{artiprinci}, which allow characterizing these elements, calculating the exact amount of k-normal elements present in a finite field and calculating their density. Finally, it is presented a result given by the authors mentioned above, which guarantees the existence of primitive 1-normal elements for certain finite fields.