Let $p$ be a prime number and $\chi$ be a primitive Dirichlet character of conductor $p$. We let $$g(\chi)=\sum_{a=1}^{p}{\chi(a)e^{2i\pi a/p}}$$ be the Gauss sum attached to $\chi$. Is this known that $g(\chi)$ does not belong to $\mathbb{Q}(\chi)$, the algebraic extension of $\mathbb{Q}$ generated by the values of $\chi$?

Many thanks!