How can we prove that every field of characteristic 0 has at least one Discrete Valuation Ring?
My effort: Let $K$ be an field of characteristic 0. Then $\mathbb{Z}$ is a subring of $K$. Let $p$ be a prime. By Theorem 10.2 in Matsumura, there exists a valuation ring $R$ of $K$ with $\mathbb{Z} \subset R$ and $m_R \cap \mathbb{Z}=p \mathbb{Z}$, where $m_R$ is the maximal ideal of $R$. If I could show that $R$ is Noetherian, or principal ideal domain, then I would be done by Theorem 11.1 of Matsumura. But I am having a hard time proving this and besides, it seems to me that this is not the right direction.
Edit: This question was motivated by the remark in Matsumura's Commutative Ring Theory p. 79, which mentions "Let $K$ be a field and $R$ a DVR of $K$..." As the answers point out, a field need not have a DVR. Then why would $K$ have a DVR in Matsumura's remark?