integrity characterized by places

Theorem.

Let $R$ be a subring of the field $K$,  $1\in R$.  An element $\alpha$ of the field is integral over $R$ if and only if all places (http://planetmath.org/PlaceOfField) $\phi$ of $K$ satisfy the implication

$$\phi \mathrm{is}\mathrm{finite}\mathrm{in}R\Rightarrow \phi (\alpha )\mathrm{is}\mathrm{finite}.$$

1.  Let $R$ be a subring of the field $K$,  $1\in R$.  The integral closure of $R$ in $K$ is the intersection of all valuation domains in $K$ which contain the ring $R$.  The integral closure is integrally closed in the field $K$.

2.  Every valuation domain is integrally closed in its field of fractions.