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) $\varphi$ of $K$ satisfy the implication

 $\varphi\mathrm{\,\,is\,finite\,in\,}R\,\,\,\Rightarrow\,\,\varphi(\alpha)% \mathrm{\,is\,finite}.$

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.

Title integrity characterized by places IntegrityCharacterizedByPlaces 2013-03-22 14:56:57 2013-03-22 14:56:57 pahio (2872) pahio (2872) 11 pahio (2872) Theorem msc 12E99 msc 13B21 Integral PlaceOfField