# place of field

Let $F$ be a field and $\infty$ an element not belonging to $F$.  The mapping

 $\varphi:\,k\to F\cup\{\infty\},$

where $k$ is a field, is called a place of the field $k$, if it satisfies the following conditions.

• The preimage$\varphi^{-1}(F)=\mathfrak{o}$  is a subring of $k$.

• The restriction$\varphi|_{\mathfrak{o}}$  is a ring homomorphism from $\mathfrak{o}$ to $F$.

• If  $\varphi(a)=\infty$,  then  $\varphi(a^{-1})=0$.

It is easy to see that the subring $\mathfrak{o}$ of the field $k$ is a valuation domain; so any place of a field determines a unique valuation domain in the field.  Conversely, every valuation domain $\mathfrak{o}$ with field of fractions $k$ determines a place of $k$:

###### Theorem.

Let $\mathfrak{o}$ be a valuation domain with field of fractions $k$ and $\mathfrak{p}$ the maximal ideal of $\mathfrak{o}$, consisting of the non-units of $\mathfrak{o}$.  Then the mapping

 $\varphi:\,k\to\mathfrak{o/p}\cup\{\infty\}$

defined by

 $\varphi(x):=\begin{cases}x+\mathfrak{p}\quad\mathrm{when}\,\,\,x\in\mathfrak{o% },\\ \infty\quad\mathrm{when}\,\,\,x\in k\smallsetminus\mathfrak{o},\end{cases}$

is a place of the field $k$.

Proof.  Apparently,  $\varphi^{-1}(\mathfrak{o/p})=\mathfrak{o}$  and the restriction  $\varphi|_{\mathfrak{o}}$  is the canonical homomorphism from the ring $\mathfrak{o}$ onto the residue-class ring $\mathfrak{o/p}$.  Moreover, if  $\varphi(x)=\infty$,  then $x$ does not belong to the valuation domain $\mathfrak{o}$ and thus the inverse element $x^{-1}$ must belong to it without being its unit.  Hence $x^{-1}$ belongs to the ideal $\mathfrak{p}$ which is the kernel of the homomorphism$\varphi|\mathfrak{o}$.  So we see that  $\varphi(x^{-1})=0$.

## References

• 1 Emil Artin: .  Lecture notes.  Mathematisches Institut, Göttingen (1959).
 Title place of field Canonical name PlaceOfField Date of creation 2013-03-22 14:56:51 Last modified on 2013-03-22 14:56:51 Owner pahio (2872) Last modified by pahio (2872) Numerical id 16 Author pahio (2872) Entry type Theorem Classification msc 13F30 Classification msc 13A18 Classification msc 12E99 Synonym place Synonym spot of field Related topic KrullValuation Related topic ValuationDeterminedByValuationDomain Related topic IntegrityCharacterizedByPlaces Related topic RamificationOfArchimedeanPlaces Defines place of field