The Stacks project

Example 10.161.2. Let $k$ be a field. The domain $R = k[x_1, x_2, x_3, \ldots ]$ is N-2, but not Noetherian. The reason is the following. Suppose that $R \subset L$ and the field $L$ is a finite extension of the fraction field of $R$. Then there exists an integer $n$ such that $L$ comes from a finite extension $L_0/k(x_1, \ldots , x_ n)$ by adjoining the (transcendental) elements $x_{n + 1}, x_{n + 2}$, etc. Let $S_0$ be the integral closure of $k[x_1, \ldots , x_ n]$ in $L_0$. By Proposition 10.162.16 below it is true that $S_0$ is finite over $k[x_1, \ldots , x_ n]$. Moreover, the integral closure of $R$ in $L$ is $S = S_0[x_{n + 1}, x_{n + 2}, \ldots ]$ (use Lemma 10.37.8) and hence finite over $R$. The same argument works for $R = \mathbf{Z}[x_1, x_2, x_3, \ldots ]$.