place of field
Let $F$ be a field and $\mathrm{\infty}$ an element not belonging to $F$. The mapping
$$\phi :k\to F\cup \{\mathrm{\infty}\},$$ 
where $k$ is a field, is called a place of the field $k$, if it satisfies the following conditions.
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The restriction^{} ${\phi }_{\U0001d52c}$ is a ring homomorphism^{} from $\U0001d52c$ to $F$.

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If $\phi (a)=\mathrm{\infty}$, then $\phi ({a}^{1})=0$.
It is easy to see that the subring $\U0001d52c$ 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 $\U0001d52c$ with field of fractions^{} $k$ determines a place of $k$:
Theorem.
Let $\U0001d52c$ be a valuation domain with field of fractions $k$ and $\U0001d52d$ the maximal ideal^{} of $\U0001d52c$, consisting of the nonunits of $\U0001d52c$. Then the mapping
$$\phi :k\to \U0001d52c/\U0001d52d\cup \{\mathrm{\infty}\}$$ 
defined by
$$\phi (x):=\{\begin{array}{cc}x+\U0001d52d\hspace{1em}\mathrm{when}x\in \U0001d52c,\hfill & \\ \mathrm{\infty}\hspace{1em}\mathrm{when}x\in k\setminus \U0001d52c,\hfill & \end{array}$$ 
is a place of the field $k$.
Proof. Apparently, ${\phi}^{1}(\U0001d52c/\U0001d52d)=\U0001d52c$ and the restriction ${\phi }_{\U0001d52c}$ is the canonical homomorphism from the ring $\U0001d52c$ onto the residueclass ring $\U0001d52c/\U0001d52d$. Moreover, if $\phi (x)=\mathrm{\infty}$, then $x$ does not belong to the valuation domain $\U0001d52c$ and thus the inverse element ${x}^{1}$ must belong to it without being its unit. Hence ${x}^{1}$ belongs to the ideal $\U0001d52d$ which is the kernel of the homomorphism^{} $\phi \U0001d52c$. So we see that $\phi ({x}^{1})=0$.
References
 1 Emil Artin: . Lecture notes. Mathematisches Institut, Göttingen (1959).
Title  place of field 
Canonical name  PlaceOfField 
Date of creation  20130322 14:56:51 
Last modified on  20130322 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 