Polinômios sobre corpos finitos

Abstract

This work aims to study polynomials over finite fields, rank them, and, according totheir order, find the amount of monic irreducible polynomials in a finite field. Since it is a literature review, we begin by recalling basic concepts of algebra necessary for its development, such as rings, polynomials, cyclic groups, among others. Initially, we present the concept of polynomials with coe_cients in a ring, how these elements behave and what their properties are. After that, we go to a more complete structure, the field, more specifically, the finite field. In this new structure, polynomials remain as the focus. We verify the existence of these polynomials’ roots as well as their order, characterizing them as irreducible or reducible. Finally, we examine the possibility of extending the field by adding new elements, namely the roots of irreducible polynomials over this field, in order to make it possible to obtain prior irreducible factors of polynomials which become reducible in the extension of the field.