The Stacks project

Proof. By the Hilbert Nullstellensatz (Theorem 10.34.1) we know the residue field $\kappa = \kappa (\mathfrak m)$ is a finite extension of $k$. Denote $\alpha _ i \in \kappa $ the image of $x_ i$. Denote $\kappa _ i = k(\alpha _1, \ldots , \alpha _ i) \subset \kappa $, $i = 1, \ldots , n$ and $\kappa _0 = k$. Note that $\kappa _ i = k[\alpha _1, \ldots , \alpha _ i]$ by field theory. Define inductively elements $f_ i \in \mathfrak m \cap k[x_1, \ldots , x_ i]$ as follows: Let $P_ i(T) \in \kappa _{i-1}[T]$ be the monic minimal polynomial of $\alpha _ i $ over $\kappa _{i-1}$. Let $Q_ i(T) \in k[x_1, \ldots , x_{i-1}][T]$ be a monic lift of $P_ i(T)$ (of the same degree). Set $f_ i = Q_ i(x_ i)$. Note that if $d_ i = \deg _ T(P_ i) = \deg _ T(Q_ i) = \deg _{x_ i}(f_ i)$ then $d_1d_2\ldots d_ i = [\kappa _ i : k]$ by Fields, Lemmas 9.7.7 and 9.9.2.

We claim that for all $i = 0, 1, \ldots , n$ there is an isomorphism

By construction the composition $k[x_1, \ldots , x_ i] \to k[x_1, \ldots , x_ n] \to \kappa $ is surjective onto $\kappa _ i$ and $f_1, \ldots , f_ i$ are in the kernel. This gives a surjective homomorphism. We prove $\psi _ i$ is injective by induction. It is clear for $i = 0$. Given the statement for $i$ we prove it for $i + 1$. The ring extension $k[x_1, \ldots , x_ i]/(f_1, \ldots , f_ i) \to k[x_1, \ldots , x_{i + 1}]/(f_1, \ldots , f_{i + 1})$ is generated by $1$ element over a field and one irreducible equation. By elementary field theory $k[x_1, \ldots , x_{i + 1}]/(f_1, \ldots , f_{i + 1})$ is a field, and hence $\psi _ i$ is injective.

This implies that $\mathfrak m = (f_1, \ldots , f_ n)$. Moreover, we also conclude that

Hence $(f_1, \ldots , f_ i)$ is a prime ideal. Thus

is a chain of primes of length $n$. The lemma follows. $\square$