I am working on a proof of the Krull dimension of $K[x_1,\dots, x_n]$ for $K$ a field. I am stuck on the following step.
Suppose that $\dim(K[x_1,\dots, x_l]) = l$ and that the maximal chains of $k[x_1,\dots, x_l]$ have length $l$ for all $l<n$. Let $P_0\subset \cdots \subset P_m$ be a chain of prime ideals in $k[x_1,\dots, x_n]$. Without loss of generality, $P_0 = 0$ and $P_1$ is a minimal nonzero prime. Show that $k[x_1,\dots, x_{n}]/P_1$ is integral over $k[x_1,\dots, x_{n-1}]$.
My guess is that we need to argue that there is an element in $P_1$ with leading term $x_n$. Then, the result would be obvious.
Also, maybe it will add some context to some of the assumptions, but the next part is to show that if $Q_i = P_i/P_1 \cap k[x_1,\dots, x_n]$, then $Q_{i}\subset Q_{i+1}$ for all $1 \leq i \leq m$.