Bounding the rational sums of squares over totally real fields

TL;DR

This paper improves bounds on expressing sums of squares over totally real Galois fields, providing a constructive proof that enhances understanding of sums of squares in algebraic number theory.

Contribution

It refines Hillar's bounds for sums of squares over totally real fields and offers a constructive proof of these improved bounds.

Findings

Improved bounds for N(m) in sums of squares over totally real fields

Constructive proof of the bounds

Enhanced understanding of sums of squares in algebraic number theory

Abstract

Let K be a totally real Galois number field. C. J. Hillar proved that if f in Q[x_1,...,x_n] is a sum of m squares in K[x_1,...,x_n], then f is a sum of N(m) squares in Q[x_1,...,x_n]. Modifying Hillar's proof, we improve the improve the bound given for N(m), the proof being constructive as well.