Sums of squares over totally real fields are rational sums of squares

Let $K$ be a totally real number field with Galois closure $L$. We prove that if $f\in \mathbb{Q}[x_1,\mathrm{.}\mathrm{.}\mathrm{.},x_n]$ is a sum of $m$ squares in $K[x_1,\mathrm{.}\mathrm{.}\mathrm{.},x_n]$, then $f$ is a sum of $$4m\cdot 2^{[L:\mathbb{Q}]+1}\left(\genfrac{}{}{0px}{}{[L:\mathbb{Q}]+1}{2}\right)$$ squares in $\mathbb{Q}[x_1,\mathrm{.}\mathrm{.}\mathrm{.},x_n]$. 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.