Topics in finite fields: Artin-Schreier's curves, superelliptic curves and irreducible polynomials

• Autor(a) principal: Daniela Alves de Oliveira
• Data de Publicação: 2023
• Tipo de documento: Tese
• Idioma: eng
• Título da fonte: Repositório Institucional da UFMG
• Texto Completo: http://hdl.handle.net/1843/51317; https://orcid.org/0000-0001-6809-6254
• topic: Finite Fields; Quadratic Forms; Artin-Schreier's Curves; Artin-Schreier's Hypersurfaces; Superelliptic Curves; Hasse-Weil's Bound; Gauss Sums; Circulant Matrices; Irreducible Polynomials; Matemática – Teses; Corpos finitos (Algebra) – Teses; Formas Quadráticas – Teses; Somas de Gauss – Teses.

In this thesis we study some problems in the finite field theory that interesting for their applications in coding theory, cryptography, communications and related areas. Our first problem is to determine the number of rational points of a family of Artin-Schreier curves and of an Artin-Schreier hypersurface, as well as to determine conditions for these curves/hypersurface to be maximal or minimal with respect to the Hasse-Weil bound. In the sequence, we study a class of superelliptic curves and, under some conditions, we describe the number of rational points of these curves. The last topic of this work is about irreducible polynomials, where we determine conditions on n and q for which the irreducible factors over F_q of the binomial x^n-1 are binomials and trinomials.