Arithmetic of equations $p=x^2+ny^2$

Abstract

In this thesis, we mainly focus on two subjects. The first is class field theory in ideal language which introduces the Artin map and guarantees the existence of class fields. The second is to find equivalence conditions which primes can be written in the form of $x^2+ny^2$. Gauss first introduced field theory beyond arithmetic approach, and we will use modern class field theory to give the exact description for the solution of this problem. Although $x^2+ny^2$ is a type of quadratic forms, we will introduce general quadratic forms, and make a class group. Then we will relate a form class group with an ideal class group, which leads us to class field theory. Our problem is closely related to a prime ideal factorization and the class fields, actually the Hilbert class fields and the ring class fields. For an example, we will give an equivalence condition for the primes expressed in the form of $x^2+64y^2$.