A Gauss sum over a field.

Question

Let $K$ be a field (not necessarily $\mathbb C$) and let $\zeta=\zeta_n$ be a primitive $n$th root of unity in $\bar K$.

I would like to know if there is a formula calculating $$ \sum_{k=1}^n \zeta^{-k^2}.$$

If it exists please tell me the formula.

Thank you in advance.