Exercise 10, § 2

(a) Let $A$ be an integral domain, $K$ its field of fractions and $m$ a positive integer. Let $t_1, ..., t_m$ be elements of $K$ and let $\varphi : A[X_1, ..., X_m] \to A[t_1, ..., t_m]$ be the unique homomorphism of $A$-algebras such that $\varphi(X_i) = t_i$. Now prove that the height of the kernel ideal of $\varphi$ is equal to $m$. (b) Deduce that for every $m$-tuple $(t_1, ..., t_m)$ of elements of $K$, one has $\dim(A[t_1, ..., t_m]) \leq \dim(A[X_1, ..., X_m]) - m$.