Existência de elementos primitivos k-normais em corpos finitos

Abstract

This dissertation investigates conditions for the existence of elements that are simultaneously primitive and k normal in finite field extensions. The study of the existence of such elements is a central problem in modern Algebra, with direct implications on the efficiency of cryptographic systems and the construction of error-correcting codes. To approach this problem, we employ a theoretical framework that combines combinatorial and analytical methods. The primary tools utilized are the Euler totient function (ϕ) and the Möbius function (µ). Within the scope of Linear Algebra over finite fields, we explore the properties of linearized polynomials, which allow the characterization of the linear independence of an element’s conjugates under the action of Frobenius automorphisms. To prove the main results, we use tools such as additive and multiplicative characters, a particular type of homomorphism, integrating them with the study of Gaussian sums, which provide the estimates for exponential sums needed to guarantee that the number of desired elements is strictly positive.