Let $K$ be a number field, and let $I$ be an ideal of the polynomial ring $K[x_1,\dotsc,x_n]$.
Assume that $K$ is the minimal field of definition of $I$.
Let $L$ be the minimal field of definition of $\sqrt{I}$.
I can prove that $L\subset K$: Take $\sigma\in\operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ such that $\sigma I=I$. Then $\sigma\sqrt{I}=\sqrt{\sigma I}=\sqrt{I}$, proving the claim.
But is $L$ always equal to $K$, or can the inclusion sometimes be proper?