If I am correct, your Noetherian local domain $(R,m(R))$ admits a place $D$ (maximal local domain) such that $R\subset D\subset K$ and $m(D)\cap R=m(R)$ (for a local ring $L$, we note $m(L)$ its maximal ideal). The fraction field $K$ is common to $R$ and $D$. As $D$ is a place there is a valuation $$ v\ :\ K\to \Gamma_{\infty} $$
such that $$ D=\{x\in K\,|\, 0\leq v(x)<\infty\}\ ;\ m(D)=\{x\in K\,|\, 0< v(x)<\infty\} $$ this valuation defines the same quotient topology ($R$ is endowed with the $m$-adic topology and $R'=R\setminus \{0\}$ with the induced topology) $$ R\times R'\to K\mbox{ and } D\times D'\to K $$ which automatically is that of a topological field, see the construction here. This topology induces the given ($m$-adic) topology on $R$.

If I am correct, your Noetherian local domain $(R,m(R))$ admits a place $D$ (maximal local domain) such that $R\subset D\subset K$ and $m(D)\cap R=m(R)$ (for a local ring $L$, we note $m(L)$ its maximal ideal). The fraction field $K$ is common to $R$ and $D$. As $D$ is a place there is a valuation $$ v\ :\ K\to \Gamma_{\infty} $$
such that $$ D=\{x\in K\,|\, 0\leq v(x)<\infty\}\ ;\ m(D)=\{x\in K\,|\, 0< v(x)<\infty\} $$ this valuation defines the same quotient topology ($R$ is endowed with the $m$-adic topology and $R'=R\setminus \{0\}$ with the induced topology) $$ R\times R'\to K\mbox{ and } D\times D'\to K $$ which automatically is that of a topological field, see the construction here. This topology induces the given ($m$-adic) topology on $R$.

If I am correct, your Noetherian local domain $(R,m(R))$ admits a place $D$ (maximal local domain) such that $R\subset D\subset K$ and $m(D)\cap R=m(R)$ (for a local ring $L$, we note $m(L)$ its maximal ideal). The fraction field $K$ is common to $R$ and $D$. As $D$ is a place there is a valuation $$ v\ :\ K\to \Gamma_{\infty} $$
such that $$ D=\{x\in K\,|\, 0\leq v(x)<\infty\}\ ;\ m(D)=\{x\in K\,|\, 0< v(x)<\infty\} $$ this valuation defines the quotient topology $$ R\times (R\setminus \{0\})\to K $$ which automatically is a topological field, see the construction here.

If I am correct, your Noetherian local domain $(R,m(R))$ admits a place $D$ (maximal local domain) such that $R\subset D\subset K$ and $m(D)\cap R=m(R)$ (for a local ring $L$, we note $m(L)$ its maximal ideal). The fraction field $K$ is common to $R$ and $D$. As $D$ is a place there is a valuation $$ v\ :\ K\to \Gamma_{\infty} $$
such that $$ D=\{x\in K\,|\, 0\leq v(x)<\infty\}\ ;\ m(D)=\{x\in K\,|\, 0< v(x)<\infty\} $$ this valuation defines the same quotient topology ($R$ is endowed with the $m$-adic topology and $R'=R\setminus \{0\}$ with the induced topology) $$ R\times R'\to K\mbox{ and } D\times D'\to K $$ which automatically is that of a topological field, see the construction here. This topology induces the given ($m$-adic) topology on $R$.