Let $k$ be a field ; let $R=k[x_1,...,x_n]$ be a field which is a finitely generated algebra over $k$ , generated by $x_1,...,x_n \in R$ . Let $S=k[x_{t+1},...,x_n]$ . If $x_{t+1},...,x_n$ are independent transcendental's over $k$ and $x_1,...,x_t$ are algebraic over $k(x_{t+1},...,x_n)$ , then does there exist $0 \ne y \in S$ such that $yx_1,...,yx_n$ are integral over $S$ ?
I need this in a proof of Zariski's lemma for Hilbert Nullstellansatz .
Please help . Thanks in advance
If $y_j\in S$ is such that $y_jx_j$ is integral over $S$ then you can take $y=y_1\cdots y_t$.
You can take a minimal polynomial for $x_j$ over $k(x_{t+1},\ldots,x_n)$ and multiply it by a common denominator to give a nonzero polynomial $\phi_j$ over $S$ with $\phi_j(x_j)=0$. Take $y_j$ to be the leading coefficient of $\phi_j$.