On the sum of two integral squares in certain quadratic fields

TL;DR

This paper establishes a precise, computable criterion based on the integral Brauer-Manin obstruction for identifying integers that can be expressed as a sum of two squares within specific quadratic fields, leveraging reciprocity laws.

Contribution

It provides a necessary and sufficient condition for sums of two squares in certain quadratic fields, using the integral Brauer-Manin obstruction, advancing the understanding of representability in algebraic number theory.

Findings

Derived a computable criterion for sums of two squares in quadratic fields

Connected the criterion to the reciprocity law

Utilized the integral Brauer-Manin obstruction for the characterization

Abstract

In this note, we give a necessary and sufficient condition for determining which integers can be written as a sum of two integral squares for certain quadratic fields by using the integral Brauer-manin obstruction (see \cite{CTX}). The condition is computable and originally from the reciprocity law.