A Short Proof for the Krull Dimension of a Polynomial Ring

A first important result in dimension theory is the fact that the Krull dimension of the ring K[X1, . . . , X`] is equal to ` when K is a field. In the literature this result is always obtained after some preliminary efforts that seem excessive for settling such an intuitive fact. For example, many authors rely on the principal ideal theorem of Krull, whose proof is very tricky. Matsumura [6,chap.2] gives a rather direct proof, but his Theorem 5.6 settling the result needs Theorems 5.1 to 5.5 together with three pages of rather subtle arguments, including Hilbert’s Nullstellensatz. The shortest proof we are aware of appears in [3]. The goal of this note is to give a short, direct proof of this fact, based on an elementary elementwise characterization of Krull dimension (Corollary 2).