Corpos finitos e dois problemas olímpicos - Nascimento, Antônio Mário Alves do (2019)
In this dissertation we present a study about abstract algebra, more precisely about finite bodies. The objective of this paper is to present the solution of the problems ”Let the positive integer and p be a prime divisor of a3− 3a + 1 with p other than 3. Prove that p is of the form 9k +1 or 9k − 1, being k integer.” Proposed in OBM 2017 Level 3 e ”Demonstrate that for each integer a> 1, the prime dividers of the number 5a4− 5a2 + 1 are of the form 20k ± 1, k ∈ Z.” Proposed at the 13th Ibero-Mathematical Olympiad American. In this sense, we begin with the introduction of group theory and present basic concepts and important theorems such as Lagrange's theorem. We then introduce the ring theory, present important definitions as quotient ring, and highlight the polynomial ring. Later we began the study of bodies. We will study body construction from an irreducible polynomial, body extension, decomposition body, and characterization of finite bodies. Finally, we provide solutions to the problems mentioned above.