Corollary 3

Let $A$ be a noetherian ring and $n$ an integer $\geqslant 0$. Then we have

Put $B = A[X_1, ..., X_n]$. For every prime ideal $p$ of $A$, the ring $B \otimes_A \kappa(p)$ is a polynomial ring in $n$ variables over a field, hence is of dimension $n$ ($§ 2$, No. 4, Cor. 1 to Theorem 3). By Cor. 2, we have $\dim(B) \leqslant \dim(A) + n$; the reverse inequality follows from Example 4 of $§ 1$, No. 3.