Representación de números primos mediante formas cuadráticas

Abstract

The well-known seventeenth century mathematician Pierre de Fermat left some interesting results on integers and how to represent them as sums of powers – all of them unproved. In this dissertation we study the following problem: given an integer n, which prime numbers p can be expressed in the form p = x 2 + ny2, where x and y are integers? We will start with some basic notions of algebraic number theory, such as the ring of integers of a number field. Then, we will move forward to unique factorization of ideals and Dedekind domains, using the class number of a number field to help us look at a specific direction: rings of algebraic integers which are principal ideal domains. We will close this dissertation by following the theory presented here and giving some examples of primes being expressed as the problem states.