Let $R$ be a commutative ring with identity. Consider the polynomial ring in $n$ indeterminates $R[X_{1},\ldots ,X_{n}]$. Let $R[Y_{1},\ldots ,Y_{m}]$ be a polynomial sub-ring of it in $m$ indeterminates. I want to prove that $m\leq n$. So I'm given two facts here:
- $Y_{i},1\leq i \leq m$ can be written in terms of $X_{j},1\leq j\leq n$.
- $\{Y_{i}\}_{1\leq i \leq m}$ are algebraically independent over $R$.
Hints are most welcome.