If $k$ is a finite field with $q$ elements and $I$ an ideal of $R=k[x_0,\dots,x_n]$, then $\overline I=\mathrm{rad}\bigl(I+\sum_{i\le n}(x_i^q-x_i)R\bigr)$. This follows immediately from Hilbert’s Nullstellensatz applied to the algebraic closure of $k$.

On an unrelated note, a more explicit description for the case of $k$ real-closed follows from Stengle’s (Positiv- and) Nullstellensatz: $f\in\overline I$ iff $-f^{2n}\in I+\Sigma$ for some $n\in\mathbb N$, where $\Sigma$ is the set of all sums of squares of polynomials.

If $k$ is a finite field with $q$ elements and $I$ an ideal of $k[x_1,\dots,x_n]$, then $\overline I=I+I_0$, where $I_0=(x_1^q-x_1,\dots,x_n^q-x_n)$. This follows immediately from Hilbert’s Nullstellensatz applied to the algebraic closure of $k$, and the observation that any ideal extending $I_0$ is a radical ideal (as it contains all polynomials of the form $f^q-f$).

On an unrelated note, a more explicit description for the case of $k$ real-closed follows from Stengle’s (Positiv- and) Nullstellensatz: $f\in\overline I$ iff $-f^{2n}\in I+\Sigma$ for some $n\in\mathbb N$, where $\Sigma$ is the set of all sums of squares of polynomials.

If $k$ is a finite field with $q$ elements and $I$ an ideal of $R=k[x_0,\dots,x_n]$, then $\overline I=\mathrm{rad}\bigl(I+\sum_{i\le n}(x_i^q-x_i)R\bigr)$. This follows immediately from Hilbert’s Nullstellensatz applied to the algebraic closure of $k$.

If $k$ is a finite field with $q$ elements and $I$ an ideal of $R=k[x_0,\dots,x_n]$, then $\overline I=\mathrm{rad}\bigl(I+\sum_{i\le n}(x_i^q-x_i)R\bigr)$. This follows immediately from Hilbert’s Nullstellensatz applied to the algebraic closure of $k$.

On an unrelated note, a more explicit description for the case of $k$ real-closed follows from Stengle’s (Positiv- and) Nullstellensatz: $f\in\overline I$ iff $-f^{2n}\in I+\Sigma$ for some $n\in\mathbb N$, where $\Sigma$ is the set of all sums of squares of polynomials.