Primes in quadratic fields

TL;DR

This paper introduces algorithms for computing quadratic characters and prime ideal norms in quadratic fields, utilizing a sieve method and geometric representations to visualize primes within the field's ring of integers.

Contribution

It presents novel algorithms for calculating quadratic characters and prime ideal norms, with a geometric approach to visualize primes in quadratic fields.

Findings

Algorithms successfully compute quadratic characters and prime norms.

Prime ideals are visualized in a plane representation of quadratic fields.

Sieve method effectively identifies prime ideals within bounded regions.

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.