Sums of Quaternion Squares and a Theorem of Watson

TL;DR

This paper investigates sums of squares in quaternion rings with integer coefficients, applying Watson's theorem to identify conditions under which elements can be expressed as the sum of three squares.

Contribution

It introduces a new application of Watson's theorem to characterize sums of squares in quaternion rings, expanding understanding of their algebraic structure.

Findings

Identifies conditions for elements in quaternion rings to be sums of three squares

Determines a large family of quaternion rings where sum-of-squares elements are expressible as three squares

Provides a framework for analyzing sums of squares in non-commutative rings

Abstract

We use a representability theorem of G. L. Watson to examine sums of squares in Quaternion rings with integer coefficients. This allows us to determine a large family of such rings where every element expressible as the sum of squares can be written as the sum of 3 squares.