There are two theorems in Matsumura (p. 78-9)
Theorem 11.1 Let $R$ be a valuation ring. Then the following conditions are equivalent:
(1) $R$ is a DVR
(2) $R$ is a PID
(3) $R$ is Noetherian
and
Theorem 11.2 Let $R$ be a ring; then the following conditions are equivalent:
(1) $R$ is a DVR
(2) $R$ is a local PID, and not a field
(3) $R$ is a Noetherian local ring, dim $R >0$ and the maximal ideal $\mathfrak{m}_R$ is principal
(4) $R$ is a one-dimensional normal Noetherian local ring.
Question
Suppose we have a field $K$. Then $K$ is trivially a valuation ring. It is also (of course) a PID, then according to Thereorem 11.1 it must be a DVR.
Since $K$ is a DVR, $K$ is not a field according to 11.2. Am I misunderstanding something?