= One-dimensional normal local rings are discrete valuation rings
Let $R$ be a one-dimensional normal Noetherian local domain. Choose $0\ne x$ in its maximal ideal $\mathfrak m$ and a minimal $n$ with $\mathfrak m^n\subseteq(x)$. If $n>1$, choose $y\in\mathfrak m^{n-1}\setminus(x)$ and put $z=y/x$. Then $z\notin R$ but $z\mathfrak m\subseteq R$. If $z\mathfrak m\subseteq\mathfrak m$, the <determinant trick> makes $z$ integral, contrary to normality. Thus $zu$ is a unit for some $u\in\mathfrak m$, and $v=u(zv)/(zu)$ for every $v\in\mathfrak m$ proves $\mathfrak m=(u)$. The case $n=1$ is already principal. Such a local domain is a <discrete valuation ring>.
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