Sobre elementos distinguidos em corpos finitos

Abstract

In this thesis, we show some results about primitive normal elements and their generalizations. The thesis begins by studying the existence of primitive normal elements whose image by a rational function is primitive. The second result draws on a work of Cohen which studies consecutive primitive elements. In the thesis, we deal with arithmetic progressions, with a given common difference, such that all elements are primitive and one of them is normal. Next, we look for explicit formulas for the number of $k$-normal elements in extensions of finite fields. In particular, we find a formula that depends on the number of solutions of certain Diophantine equations. For k=0,1,2,3 we find combinatorial formulas that are easy to compute. We also study the existence of 2-normal primitive elements and finally we study r-primitive k-normal elements in general, and apply these results to the particular case where the field has characteristic 11, r=3 and k=3.