Uma introdução aos polinômios sobre corpos finitos

Abstract

In this work, we study topics in finite field theory. Initially, we explore the structure of finite fields, covering from basic properties to representations of their elements, such as polynomial, cyclotomic, and matrix representations. Next, we focus on the study of polynomials over finite fields, highlighting: the order-defined polynomials, linked to the structure of multiplicative groups; the primitive polynomials; and the irreducible polynomials, fundamental for factorization and construction of field extensions. Finally, we examine q-polynomials, which generalize classical polynomial structures.