Is every local domain dominated by some Noetherian local domain?

Question

I have known that every Noetherian local domain can be dominated by a DVR in its own fraction field. And there exist some fields without any discrete valuation. So I wonder if every local domain can be dominated by a DVR in a larger field? I've searched for a long time but not got any clue at all. Thank you for your answers.

Answer 1

The answer is no. Take a valuation ring $R$ (not necessarily noetherian). If $S$ is a DVR dominating $R$, then ${\rm Quot}(R)^\times/R^\times$ embeds into ${\rm Quot}(S)^\times/S^\times\cong\mathbb{Z}$, hence is itself isomorphic to $\mathbb{Z}$, but that means that $R$ is DVR itself. In the language of valuation theory, $R$ corresponds to a valuation $v:{\rm Quot}(R)^\times\rightarrow\Gamma$ for some ordered abelian group $\Gamma$, and if $S$ dominates $R$, then $S$ is the valuation ring of some extension of $v$ to ${\rm Quot}(S)$.