Krull-dimension of the power series ring over a nondiscrete valuation domain is uncountable

Let V be a rank-one nondiscrete valuation domain with maximal ideal M. We prove that the Krull-dimension of V〚X〛V∖(0) is uncountable, and hence the Krull-dimension of V〚X〛 is uncountable. This corresponds to the well-known fact that the Krull-dimension of the ring of entire functions is uncountable. In fact we construct an uncountable chain of prime ideals inside M〚X〛 such that all the members contract to (0) in V. Our method provides a new proof that the Krull-dimension of the ring of entire functions is uncountable. It is also shown that V〚X〛V∖(0) is not even a Prüfer domain, while the ring of entire functions is a Bezout domain. These are answers to Eakin and Sathayeʼs questions. Applying the above results, we show that the Krull-dimension of V〚X〛 is uncountable if V is a nondiscrete valuation domain.