2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/167860Let $K$ be a totally real number field with Galois closure $L$. We prove that if $f \in \mathbb Q[x_1,...,x_n]$ is a sum of $m$ squares in $K[x_1,...,x_n]$, then $f$ is a sum of \[4m \cdot 2^{[L: \mathbb Q]+1} {[L: \mathbb Q] +1 \choose 2}\] squares in $\mathbb Q[x_1,...,x_n]$. Moreover, our argument is constructive and generalizes to the case of commutative $K$-algebras. This result gives a partial resolution to a question of Sturmfels on the algebraic degree of certain semidefinite programing problems.10 pages, final version to appear in Proceedings of the AMSCommutative AlgebraOptimization and ControlRings and Algebrastext