Sums of squares of integer-multiple of an integral element on real bi-quadratic fields

TL;DR

This paper investigates conditions under which multiples of certain algebraic integers in real bi-quadratic fields cannot be expressed as sums of squares, providing theoretical results and illustrative examples.

Contribution

It constructs specific algebraic integers in real bi-quadratic fields and establishes necessary conditions preventing their multiples from being sums of squares.

Findings

Identifies conditions for non-representability as sums of squares

Provides explicit examples in tabular form

Distinguishes cases based on subfield membership

Abstract

For any given positive integer $m$ we construct certain totally positive algebraic integers $\alpha$ of a real bi-quadratic field $K$ and obtain some necessary conditions for which $m\alpha$ can not be represented as sum of integral squares. We show this for integers lie in quadratic subfields of $K$ and for integers which are in $K$ but not in any quadratic subfield of $K$. We provide examples in tabular form for each cases to corroborate the results.