On the sum of two integral squares in quadratic fields $\Q(\sqrt{\pm p})

TL;DR

This paper introduces a method to identify integers that can be expressed as a sum of two integral squares within quadratic fields of the form (  p), focusing on prime p, expanding understanding of sum-of-two-squares representations in these fields.

Contribution

It presents a novel approach for determining which integers are sums of two squares in quadratic fields (  p), a problem not fully addressed before.

Findings

Method effectively identifies sum-of-two-squares integers in quadratic fields

Characterizes conditions under which integers are representable as sums of two squares

Provides new insights into quadratic field arithmetic and sum representations

Abstract

We propose a method for determining which integers can be written as a sum of two integral squares for quadratic fields $\Q(\sqrt{\pm p})$, where $p$ is a prime.