Let $k$ be a perfect field and $\bar{k}$ its algebraic closure. Let $H$ be a normal subgroup of $Gal(\bar{k}/k)$ of finite index. Let $L$ be the subfield of $\bar{k}$ of all elements which are fixed under the action of $H$. Then $L/k$ is a finite Galois extension.
Let $I$ be an ideal of $\bar{k}[X_1,\cdots,X_n]$ such that $\sigma I=I$ for every $\sigma\in H$. Is it then true that $I$ is generated as an ideal of $\bar{k}[X_1,\cdots,X_n]$ by $I\cap L[X_1,\cdots,X_n]$?
(in other words, is $I$ generated by polynomials over $L$?)
There's relevant information in What is the field of definition of an invariant ideal? , but the answers there either reference to an old book or use nonabelian cohomology. I'm looking for a more accessible proof, and hopefully not just a reference, but a proof or at least an overview of one.
Special case: Let $X$ be an affine subvariety of $\mathbb{A}^n_{\bar{k}}$ defined by polynomials over $k$ such that each of its geometric components is mapped onto itself under the action of $H$. Is it then true that each component is defined by polynomials over $L$? This is my main interest.