Krull dimension of polynomial ring in countably many variables

Question

How can I prove that the Krull dimension of the polynomial ring $R=K[X_1,X_2,...]$ in countably many variables ($K$ a field) is infinity ? I have already proved that $R$ is an integral domain but not Noetherian, because $(X_1)\subsetneq(X_1,X_2)\subsetneq\cdots$ is a chain of ideals that does not stabilise at some point. I know that the Krull dimension is the supremum of the heights of all prime ideals of the ring and that the Krull dimension of the polynomial ring $K[X_1,...,X_n]$ in finitely many variables is equal to $n$. How can I efficiently apply this knowledge in order to compute the Krull dimension for countably many variables ? Thanks for your help !

Answer 1

Hint:

For any $n\in\mathbf N$, there exists a prime ideal of height $n$. Namely, $$(X_1, X_2,\dots, X_n)\varsupsetneq(X_1, X_2,\dots, X_{n-1})\varsupsetneq\dots\varsupsetneq(X_1, X_2)\varsupsetneq (X_1)\varsupsetneq\{0\}$$ is a chain of prime ideals with length $n$.