Sums of squares over totally real fields are rational sums of squares
Christopher J. Hillar · Proceedings of the American Mathematical Society · 2008
Let $K$ be a totally real number field with Galois closure $L$. We prove that if $f \in \mathbb Q[x_1,\ldots ,x_n]$ is a sum of $m$ squares in $K[x_1,\ldots ,x_n]$, then $f$ is a sum of \[ 4m \cdot 2^{[L: \mathbb Q]+1} {[L: \mathbb Q] +1 \choose 2}\] squares in $\mathbb Q[x_1,\ldots ,x_n]$. 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 programming problems.