| Most problems in number theory are exceedingly simple to state, yet many continue to elude mathematicians even centuries after they were originally posed. Such a question, "Given a positive integer n, when can a prime number be written in the form x 2 + ny2?", was solved by Cox, and although the statement is elementary, the solution requires the depth and power of class field theory to understand. In our approach to this question, we will explore a variety of topics, including: algebraic number fields; types of class groups and class fields; two density theorems; the main theorems in class field theory; and the theory of quadratic forms. Our discussion will culminate in Theorem 2.11.3, a full characterization of primes of the form x2 + ny2.;However, the intrigue doesn't end there. In Chapter 3, we pose the related question: "If p is a prime of the form x2 + ny2, when is y2+nx2 also prime?" This question turns out to be much harder to approach, but we will investigate the symmetric n-Fermat prime question thoroughly. |
