Consider some algebraically independent polynomials $f\_1,\ldots, f\_n\in\mathbb{C}[x\_1,\ldots, x\_n]$.


Is it possible that $I\subseteq\mathbb{C}[f\_1,\ldots, f\_n]\subsetneq\mathbb{C}[x\_1,\ldots, x\_n]$ for some not trivial ideal $I$ of $\mathbb{C}[x\_1,\ldots, x\_n]$?



Big polynomial subalgebra of polynomials
It is not possible.


First, note that in the situation, $0\neq I\subset A=\mathbb{C}[f\_1,\ldots, f\_n]\subset B=\mathbb{C}[x\_1,\ldots, x\_n]$ with $I$ an ideal of $B$, one must have $A$ and $B$ birational. To see this, let $0\neq p\in I$. Then, $x\_ip\in I$ and thus, $x\_i=x\_ip/p$.


Next, assume that $x\_i\not\in A$. Write $x\_i=F/G$ with $F,G\in A$ and coprime in $A$, since $A$ is a polynomial ring and thus a UFD. Then, $x\_i^mp\in I$ and so, $F^mp/G^m\in A$. This says, $G^m$ divides $p$ in $A$ for all $m$ and the only way this can happen is if $G=1$ and so $x\_i\in A$.


