Primes in quadratic fields

Abstract

This paper presents algorithms for calculating the quadratic character and the norms of prime ideals in the ring of integers of any quadratic field. The norms of prime ideals are obtained by means of a sieve algorithm using the quadratic character for the field considered. A quadratic field, and its ring of integers, can be represented naturally in a plane. Using such a representation, the prime numbers - which generate the principal prime ideals in the ring - are displayed in a given bounded region of the plane.