On calculating the restriction of the ideals in polynomial rings.

Question

Let $K$ be a field and $F$ a Galois extension of $K$ and $G$ the Galois group of extension $F/K$. For any $\sigma\in G$, define an $K$-algebra automorphism $\sigma^*$ on $F[X_1,\dots,X_n]$ by $X_i\mapsto X_i, a\mapsto\sigma a\ (\forall a\in F$), then $\sigma\mapsto\sigma^*$ defines an action of $G$ on $F[X_1,\dots,X_n]$.

Now consider the embedding $i:K[X_1,\dots,X_n]\to F[X_1,\dots,X_n]$. My question is if for any ideal $I$ of $F[X_1,\dots,X_n]$, we have

$$F[X_1,\dots,X_n]i(i^{-1}I)=\bigcap_{\sigma\in G}\sigma^*I.$$

Answer 1

Let $I = (X-i,Y-i)(X+i,Y+i) \subset \mathbb{C}[X,Y]$.

Then $f=(X-i)(Y+i)$ and its complex conjugate are both in $I$, but $f$ is not generated by $I\cap \mathbb{R}[X,Y] = (X^2+1,Y^2+1,XY+1)$.